THE ASTROPHYSICAL JOURNAL, 517 : 565È586, 1999 June 1
( 1999. The American Astronomical Society. All rights reserved. Printed in U.S.A.
MEASUREMENTS OF ) AND " FROM 42 HIGH-REDSHIFT SUPERNOVAE
S. PERLMUTTER,1 G. ALDERING, G. GOLDHABER,1 R. A. KNOP, P. NUGENT, P. G. CASTRO,2 S. DEUSTUA, S. FABBRO,3
A. GOOBAR,4 D. E. GROOM, I. M. HOOK,5 A. G. KIM,1,6 M. Y. KIM, J. C. LEE,7 N. J. NUNES,2 R. PAIN,3
C. R. PENNYPACKER,8 AND R. QUIMBY
Institute for Nuclear and Particle Astrophysics, E. O. Lawrence Berkeley National Laboratory, Berkeley, CA 94720
C. LIDMAN
European Southern Observatory, La Silla, Chile
R. S. ELLIS, M. IRWIN, AND R. G. MCMAHON
Institute of Astronomy, Cambridge, England, UK
P. RUIZ-LAPUENTE
Department of Astronomy, University of Barcelona, Barcelona, Spain
N. WALTON
Isaac Newton Group, La Palma, Spain
B. SCHAEFER
Department of Astronomy, Yale University, New Haven, CT
B. J. BOYLE
Anglo-Australian Observatory, Sydney, Australia
A. V FILIPPENKO AND T. MATHESON
Department of Astronomy, University of California, Berkeley, CA
A. S. FRUCHTER AND N. PANAGIA9
Space Telescope Science Institute, Baltimore, MD
H. J. M. NEWBERG
Fermi National Laboratory, Batavia, IL
AND
W. J. COUCH
University of New South Wales, Sydney, Australia
(THE SUPERNOVA COSMOLOGY PROJECT)
Received 1998 September 8 ; accepted 1998 December 17
ABSTRACT
We report measurements of the mass density, ) , and cosmological-constant energy density, ) , of
M
"
the universe based on the analysis of 42 type Ia supernovae
discovered by the Supernova Cosmology
Project. The magnitude-redshift data for these supernovae, at redshifts between 0.18 and 0.83, are Ðtted
jointly with a set of supernovae from the Calan/Tololo Supernova Survey, at redshifts below 0.1, to yield
values for the cosmological parameters. All supernova peak magnitudes are standardized using a SN Ia
light-curve width-luminosity relation. The measurement yields a joint probability distribution of the
cosmological parameters that is approximated by the relation 0.8) [ 0.6) B [0.2 ^ 0.1 in the region
" `0.09 (1 p statistical) `0.05
of interest () [ 1.5). For a Ñat () ] ) \ 1) cosmology we Ðnd M)flat \ 0.28
M
M
"
M
~0.04
(identiÐed systematics). The data are strongly inconsistent with a " \ 0 Ñat~0.08
cosmology, the simplest
inÑationary universe model. An open, " \ 0 cosmology also does not Ðt the data well : the data indicate
that the cosmological constant is nonzero and positive, with a conÐdence of P(" [ 0) \ 99%, including
the identiÐed systematic uncertainties. The best-Ðt age of the universe relative to the Hubble time is
tflat \ 14.9`1.4(0.63/h) Gyr for a Ñat cosmology. The size of our sample allows us to perform a variety of
0
~1.1
statistical
tests to check for possible systematic errors and biases. We Ðnd no signiÐcant di†erences in
either the host reddening distribution or Malmquist bias between the low-redshift Calan/Tololo sample
and our high-redshift sample. Excluding those few supernovae that are outliers in color excess or Ðt
residual does not signiÐcantly change the results. The conclusions are also robust whether or not a
width-luminosity relation is used to standardize the supernova peak magnitudes. We discuss and constrain, where possible, hypothetical alternatives to a cosmological constant.
Subject headings : cosmology : observations È distance scale È supernovae : general
1
2
3
4
5
6
7
8
9
Center for Particle Astrophysics, University of California, Berkeley, California.
Instituto Superior Tecnico, Lisbon, Portugal.
LPNHE, CNRS-IN2P3, and University of Paris VI and VII, Paris, France.
Department of Physics, University of Stockholm, Stockholm, Sweden.
European Southern Observatory, Munich, Germany.
PCC, CNRS-IN2P3, and Collège de France, Paris, France.
Institute of Astronomy, Cambridge, England, UK.
Space Sciences Laboratory, University of California, Berkeley, California.
Space Sciences Department, European Space Agency.
565
566
PERLMUTTER ET AL.
1.
INTRODUCTION
Since the earliest studies of supernovae, it has been suggested that these luminous events might be used as standard
candles for cosmological measurements (Baade 1938). At
closer distances they could be used to measure the Hubble
constant if an absolute distance scale or magnitude scale
could be established, while at higher redshifts they could
determine the deceleration parameter (Tammann 1979 ;
Colgate 1979). The Hubble constant measurement became
a realistic possibility in the 1980s, when the more homogeneous subclass of type Ia supernovae (SNe Ia) was identiÐed
(see Branch 1998). Attempts to measure the deceleration
parameter, however, were stymied for lack of high-redshift
supernovae. Even after an impressive multiyear e†ort by
NÔrgaard-Nielsen et al. (1989), it was only possible to
follow one SN Ia, at z \ 0.31, discovered 18 days past its
peak brightness.
The Supernova Cosmology Project was started in 1988 to
address this problem. The primary goal of the project is the
determination of the cosmological parameters of the universe using the magnitude-redshift relation of type Ia supernovae. In particular, Goobar & Perlmutter (1995) showed
the possibility of separating the relative contributions of the
mass density, ) , and the cosmological constant, ", to
M
changes in the expansion
rate by studying supernovae at a
range of redshifts. The Project developed techniques,
including instrumentation, analysis, and observing strategies, that make it possible to systematically study highredshift supernovae (Perlmutter et al. 1995a). As of 1998
March, more than 75 type Ia supernovae at redshifts
z \ 0.18È0.86 have been discovered and studied by the
Supernova Cosmology Project (Perlmutter et al. 1995b,
1996, 1997a, 1997b, 1997c, 1997d, 1998a).
A Ðrst presentation of analysis techniques, identiÐcation
of possible sources of statistical and systematic errors, and
Ðrst results based on seven of these supernovae at redshifts
z D 0.4 were given in Perlmutter et al. (1997e ; hereafter
referred to as P97). These Ðrst results yielded a conÐdence
region that was suggestive of a Ñat, " \ 0 universe but with
a large range of uncertainty. Perlmutter et al. (1998b) added
a z \ 0.83 SN Ia to this sample, with observations from the
Hubble Space T elescope (HST ) and Keck 10 m telescope,
providing the Ðrst demonstration of the method of separating ) and " contributions. This analysis o†ered preliminary Mevidence for a lowÈmass-density universe with a
best-Ðt value of ) \ 0.2 ^ 0.4, assuming " \ 0. IndepenM
dent work by Garnavich
et al. (1998a), based on three
supernovae at z D 0.5 and one at z \ 0.97, also suggested a
low mass density, with best-Ðt ) \ [0.1 ^ 0.5 for " \ 0.
M a preliminary analysis
Perlmutter et al. 1997f presented
of 33 additional high-redshift supernovae, which gave a
conÐdence region indicating an accelerating universe and
barely including a low-mass " \ 0 cosmology. Recent independent work of Riess et al. (1998), based on 10 highredshift supernovae added to the Garnavich et al. (1998a)
set, reached the same conclusion. Here we report on the
complete analysis of 42 supernovae from the Supernova
Cosmology Project, including the reanalysis of our previously reported supernovae with improved calibration
data and improved photometric and spectroscopic SN Ia
templates.
2.
BASIC DATA AND PROCEDURES
The new supernovae in this sample of 42 were all dis-
Vol. 517
covered while still brightening, using the Cerro Tololo
Inter-American Observatory (CTIO) 4 m telescope with the
20482 pixel prime-focus CCD camera or the 4 ] 20482 pixel
Big Throughput Camera.10 The supernovae were followed
with photometry over the peak of their light curves and
approximately 2È3 months further (D40È60 days rest
frame) using the CTIO 4 m, Wisconsin-Indiana-YaleNOAO (WIYN) 3.6 m, ESO 3.6 m, Isaac Newton Telescope
(INT) 2.5 m, and the William Herschel Telescope (WHT) 4.2
m telescopes. (SN 1997ap and other 1998 supernovae have
also been followed with HST photometry.) The supernova
redshifts and spectral identiÐcations were obtained using
the Keck I and II 10 m telescopes with the Low-Resolution
Imaging Spectrograph (Oke et al. 1995) and the ESO 3.6 m
telescope. The photometry coverage was most complete in
Kron-Cousins R-band, with Kron-Cousins I-band photometry coverage ranging from two or three points near peak
to relatively complete coverage paralleling the R-band
observations.
Almost all of the new supernovae were observed spectroscopically. The conÐdence of the type Ia classiÐcations
based on these spectra taken together with the observed
light curves, ranged from ““ deÐnite ÏÏ (when Si II features
were visible) to ““ likely ÏÏ (when the features were consistent
with type Ia and inconsistent with most other types). The
lower conÐdence identiÐcations were primarily due to hostgalaxy contamination of the spectra. Fewer than 10% of the
original sample of supernova candidates from which these
SNe Ia were selected were conÐrmed to be nonÈtype Ia, i.e.,
being active galactic nuclei or belonging to another SN
subclass ; almost all of these nonÈSNe Ia could also have
been identiÐed by their light curves and/or position far from
the SN Ia Hubble line. Whenever possible, the redshifts
were measured from the narrow host-galaxy lines rather
than the broader supernova lines. The light curves and
several spectra are shown in Perlmutter et al. (1997e, 1997f,
1998b) ; complete catalogs and detailed discussions of the
photometry and spectroscopy for these supernovae will be
presented in forthcoming papers.
The photometric reduction and the analyses of the light
curves followed the procedures described in P97. The supernovae were observed with the Kron-Cousins Ðlter that best
matched the rest-frame B and V Ðlters at the supernovaÏs
redshift, and any remaining mismatch of wavelength coverage was corrected by calculating the expected photometric
di†erenceÈthe ““ cross-Ðlter K-correction ÏÏÈusing template
SN Ia spectra as in Kim, Goobar, & Perlmutter (1996). We
have now recalculated these K-corrections (see Nugent et
al. 1998) using improved template spectra, based on an
extensive database of low-redshift SN Ia spectra recently
made available from the Calan/Tololo survey (Phillips et al.
1999). Where available, IUE and HST spectra (Cappellaro,
Turatto, & Fernley 1995 ; Kirshner et al. 1993) were also
added to the SN Ia spectra, including those published previously (1972E, 1981B, 1986G, 1990N, 1991T, 1992A, and
1994D : in Kirshner & Oke 1975 ; Branch et al. 1993 ; Phillips et al. 1987 ; Je†ery et al. 1992 ; Meikle et al. 1996 ; Patat
et al. 1996). In Nugent et al. (1998) we show that the Kcorrections can be calculated accurately for a given day on
the supernova light curve and for a given supernova light-
10 Big Throughput Camera information is provided by G. Bernstein &
J. A. Tyson, 1998, at http ://www.astro.lsa.umich.edu/btc/user.html.
No. 2, 1999
) AND " FROM 42 HIGH-REDSHIFT SUPERNOVAE
curve width from the color of the supernova on that day.
(Such a calculation of K-correction based on supernova
color will also automatically account for any modiÐcation
of the K-correction due to reddening of the supernova ; see
Nugent et al. 1998. In the case of insigniÐcant reddening the
SN Ia template color curves can be used.) We Ðnd that these
calculations are robust to mis-estimations of the light-curve
width or day on the light curve, giving results correct to
within 0.01 mag for light-curveÈwidth errors of ^25% or
light-curve phase errors of ^5 days even at redshifts where
Ðlter matching is the worst. Given small additional uncertainties in the colors of supernovae, we take an overall systematic uncertainty of 0.02 mag for the K-correction.
The improved K-corrections have been recalculated for
all the supernovae used in this paper, including those previously analyzed and published. Several of the low-redshift
supernovae from the Calan/Tololo survey have relatively
large changes (as much as 0.1 mag) at times in their Kcorrected light curves. (These and other low-redshift supernovae with new K-corrections are used by several
independent groups in constructing SN Ia light-curve templates, so the templates must be updated accordingly.) The
K-corrections for several of the high-redshift supernovae
analyzed in P97 have also changed by small amounts at the
light-curve peak [*K(t \ 0) [ 0.02 mag] and somewhat
larger amounts by 20 days past peak [*K(t \ 20) [ 0.1
mag] ; this primarily a†ects the measurement of the restframe light-curve width. These K-correction changes
balance out among the P97 supernovae, so the Ðnal results
for these supernovae do not change signiÐcantly. (As we
discuss below, however, the much larger current data set
does a†ect the interpretation of these results.)
As in P97, the peak magnitudes have been corrected for
the light-curve width-luminosity relation of SNe Ia :
mcorr \ m ] * (s) ,
(1)
B
B
corr
where the correction term *
is a simple monotonic function of the ““ stretch factor,ÏÏ s,corr
that stretches or contracts the
time axis of a template SN Ia light curve to best Ðt the
observed light curve for each supernova (see P97 ; Perlmutter et al. 1995a ; Kim et al. 1999 ; Goldhaber et al. 1999 ; and
see Phillips 1993 ; Riess, Press, & Kirshner 1995, 1996
[hereafter RPK96]). A similar relation corrects the V -band
light curve, with the same stretch factor in both bands. For
the supernovae discussed in this paper, the template must
be time-dilated by a factor 1 ] z before Ðtting to the
observed light curves to account for the cosmological
lengthening of the supernova timescale (Goldhaber et al.
1995 ; Leibundgut et al. 1996a ; Riess et al. 1997a). P97 calculated * (s) by translating from s to *m (both describcorr
15 then using the
ing the timescale
of the supernova event) and
relation between *m and luminosity as determined by
Hamuy et al. (1995). 15
The light curves of the Calan/Tololo
supernovae have since been published, and we have directly
Ðtted each light curve with the stretched template method
to determine its stretch factor s. In this paper, for the lightcurve width-luminosity relation, we therefore directly use
the functional form
* (s) \ a(s [ 1)
(2)
corr
and determine a simultaneously with our determination of
the cosmological parameters. With this functional form, the
supernova peak apparent magnitudes are thus all
567
““ corrected ÏÏ as they would appear if the supernovae had the
light-curve width of the template, s \ 1.
We use analysis procedures that are designed to be as
similar as possible for the low- and high-redshift data sets.
Occasionally, this requires not using all of the data available at low redshift, when the corresponding data are not
accessible at high redshift. For example, the low-redshift
supernova light curves can often be followed with photometry for many months with high signal-to-noise ratios,
whereas the high-redshift supernova observations are generally only practical for approximately 60 rest-frame days
past maximum light. This period is also the phase of the
low-redshift SN Ia light curves that is Ðtted best by the
stretched-template method and that best predicts the luminosity of the supernova at maximum. We therefore Ðtted
only this period for the light curves of the low-redshift
supernovae. Similarly, at high redshift the rest-frame
B-band photometry is usually much more densely sampled
in time than the rest-frame V -band data, so we use the
stretch factor that best Ðts the rest-frame B-band data for
both low- and high-redshift supernovae, even though at
low-redshift the V -band photometry is equally well
sampled.
Each supernova peak magnitude was also corrected for
Galactic extinction, A , using the extinction law of Cardelli,
Clayton, & Mathis R(1989), Ðrst using the color excess,
E(B[V )
, at the supernovaÏs Galactic coordinates provided bySFÔD
Schlegel, Finkbeiner, & Davis (1998) and thenÈ
for comparisonÈusing the E(B[V )
value provided by
BÔHcommunication ; see
D. Burstein & C. Heiles (1998, private
also Burstein & Heiles 1982). Galactic extinction, A , was
R
calculated from E(B[V ) using a value of the total-to-selective extinction ratio, R 4 A /E(B[V ), speciÐc to each
R
supernova. These were Rcalculated
using the appropriate
redshifted supernova spectrum as it would appear through
an R-band Ðlter. These values of R range from 2.56 at
R supernova colors
z \ 0 to 4.88 at z \ 0.83. The observed
were similarly corrected for Galactic extinction. Any extinction in the supernovaÏs host galaxy or between galaxies was
not corrected for at this stage but will be analyzed separately in ° 4.
All the same corrections for width-luminosity relation,
K-corrections, and extinction (but using R \ 4.14) were
B
applied to the photometry of 18 low-redshift
SNe Ia
(z ¹ 0.1) from the Calan/Tololo supernova survey (Hamuy
et al. 1996) that were discovered earlier than 5 days after
peak. The light curves of these 18 supernovae have all been
reÐtted since P97, using the more recently available photometry (Hamuy et al. 1996) and our K-corrections.
Figures 1 and 2a show the Hubble diagram of e†ective
rest-frame B magnitude corrected for the width-luminosity
relation,
meff \ m ] * [ K [ A ,
(3)
B
R
corr
BR
R
as a function of redshift for the 42 Supernova Cosmology
Project high-redshift supernovae, along with the 18
Calan/Tololo low-redshift supernovae. (Here K
is the
cross-Ðlter K-correction from observed R bandBRto restframe B band.) Tables 1 and 2 give the corresponding IAU
names, redshifts, magnitudes, corrected magnitudes, and
their respective uncertainties. As in P97, the inner error bars
in Figures 1 and 2 represent the photometric uncertainty,
while the outer error bars add in quadrature 0.17 mag of
intrinsic dispersion of SN Ia magnitudes that remain after
568
PERLMUTTER ET AL.
Vol. 517
FIG. 1.ÈHubble diagram for 42 high-redshift type Ia supernovae from the Supernova Cosmology Project and 18 low-redshift type Ia supernovae from the
Calan/Tololo Supernova Survey after correcting both sets for the SN Ia light-curve width-luminosity relation. The inner error bars show the uncertainty due
to measurement errors, while the outer error bars show the total uncertainty when the intrinsic luminosity dispersion, 0.17 mag, of light-curveÈwidthcorrected type Ia supernovae is added in quadrature. The unÐlled circles indicate supernovae not included in Ðt C. The horizontal error bars represent the
assigned peculiar velocity uncertainty of 300 km s~1. The solid curves are the theoretical meff(z) for a range of cosmological models with zero cosmological
constant : () , ) ) \ (0, 0) on top, (1, 0) in middle, and (2, 0) on bottom. The dashed curves Bare for a range of Ñat cosmological models : () , ) ) \ (0, 1) on
" from top, (1, 0) third from top, and (1.5, [0.5) on bottom.
M "
top, (0.5, 0.5)Msecond
applying the width-luminosity correction. For these plots,
the slope of the width-brightness relation was taken to be
a \ 0.6, the best-Ðt value of Ðt C discussed below. (Since
both the low- and high-redshift supernova light-curve
widths are clustered rather closely around s \ 1, as shown
in Fig. 4, the exact choice of a does not change the Hubble
diagram signiÐcantly.) The theoretical curves for a universe
with no cosmological constant are shown as solid lines for a
range of mass density, ) \ 0, 1, 2. The dashed lines
M
represent alternative Ñat cosmologies,
for which the total
mass energy density ) ] ) \ 1 (where ) 4 "/3H2).
M are
" for () , ) ) \
" (0, 1), (0.5,
0
The range of models shown
M
"
0.5), (1, 0), which is covered by the matching solid line, and
(1.5, [0.5).
3.
FITS TO )M AND )"
The combined low- and high-redshift supernova data sets
of Figure 1 are Ðtted to the Friedman-Robertson-Walker
(FRW) magnitude-redshift relation, expressed as in P97 :
meff 4 m ] a(s [ 1) [ K [ A
B
R
BR
R
\ M ] 5 log D (z ; ) , ) ) ,
(4)
B
L
M "
where D 4 H d is the ““ Hubble-constantÈfree ÏÏ lumiL
0 and
L M 4 M [ 5 log H ] 25 is the
nosity distance
B absolute 0magnitude at
““ Hubble-constantÈfree ÏÏ BB-band
maximum of a SN Ia with width s \ 1. (These quantities
are, respectively, calculated from theory or Ðtted from
apparent magnitudes and redshifts, both without any need
for H . The cosmological-parameter results are thus also
0
completely
independent of H .) The details of the Ðtting
procedure as presented in P970 were followed, except that
both the low- and high-redshift supernovae were Ðtted
simultaneously, so that M and a, the slope of the widthluminosity relation, could Balso be Ðtted in addition to the
cosmological parameters ) and ) . For most of the
M a are statistical
"
analyses in this paper, M and
““ nuisance ÏÏ
B
parameters ; we calculate two-dimensional conÐdence
regions and single-parameter uncertainties for the cosmological parameters by integrating over these parameters, i.e.,
P() , ) ) \ // P() , ) , M , a)dM da.
M "correlations
B
Bbetween the photoAsM in "P97, the small
metric uncertainties of the high-redshift supernovae, due to
shared calibration data, have been accounted for by Ðtting
with a correlation matrix of uncertainties.11 The lowredshift supernova photometry is more likely to be uncorrelated in its calibration, since these supernovae were not
discovered in batches. However, we take a 0.01 mag systematic uncertainty in the comparison of the low-redshift
B-band photometry and the high-redshift R-band photometry. The stretch-factor uncertainty is propagated with a
Ðxed width-luminosity slope (taken from the low-redshift
11 The data are available at http ://www-supernova.lbl.gov.
No. 2, 1999
) AND " FROM 42 HIGH-REDSHIFT SUPERNOVAE
569
FIG. 2.È(a) Hubble diagram for 42 high-redshift type Ia supernovae from the Supernova Cosmology Project and 18 low-redshift type Ia supernovae from
the Calan/Tololo Supernova Survey, plotted on a linear redshift scale to display details at high redshift. The symbols and curves are as in Fig. 1.
(b) Magnitude residuals from the best-Ðt Ñat cosmology for the Ðt C supernova subset, () , ) ) \ (0.28, 0.72). The dashed curves are for a range of Ñat
M "from bottom, and (1, 0) is the solid curve on bottom. The
cosmological models : () , ) ) \ (0, 1) on top, (0.5, 0.5) third from bottom, (0.75, 0.25) second
M , ") ) \ (0, 0). Note that this plot is practically identical to the magnitude residual plot for the best-Ðt unconstrained cosmology
middle solid curve is for ()
M "1.32). (c) Uncertainty-normalized residuals from the best-Ðt Ñat cosmology for the Ðt C supernova subset, () , ) ) \
of Ðt C, with () , ) ) \ (0.73,
M "
M "
(0.28, 0.72).
supernovae ; cf. P97) and checked for consistency after the
Ðt.
We have compared the results of Bayesian and classical,
““ frequentist,ÏÏ Ðtting procedures. For the Bayesian Ðts, we
have assumed a ““ prior ÏÏ probability distribution that has
zero probability for ) \ 0 but otherwise has uniform
M
probability in the four parameters ) , ) , a, and M . For
"
B
the frequentist Ðts, we have followedMthe classical
statistical
procedures described by Feldman & Cousins (1998) to
guarantee frequentist coverage of our conÐdence regions in
the physically allowed part of parameter space. Note that
throughout the previous cosmology literature, completely
570
PERLMUTTER ET AL.
Vol. 517
TABLE 1
SCP SNE IA DATA
SN
(1)
z
(2)
p
z
(3)
mpeak
X
(4)
ppeak
X
(5)
A
X
(6)
K
BX
(7)
mpeak
B
(8)
meff
B
(9)
p eff
mB
(10)
1992bi . . . . . . .
1994F . . . . . . .
1994G . . . . . . .
1994H . . . . . . .
1994al . . . . . . .
1994am . . . . . .
1994an . . . . . .
1995aq . . . . . .
1995ar . . . . . . .
1995as . . . . . . .
1995at . . . . . . .
1995aw . . . . . .
1995ax . . . . . .
1995ay . . . . . .
1995az . . . . . . .
1995ba . . . . . .
1996cf . . . . . . .
1996cg . . . . . . .
1996ci . . . . . . .
1996ck . . . . . .
1996cl . . . . . . .
1996cm . . . . . .
1996cn . . . . . .
1997F . . . . . . .
1997G . . . . . . .
1997H . . . . . . .
1997I . . . . . . . .
1997J . . . . . . . .
1997K . . . . . . .
1997L . . . . . . .
1997N . . . . . . .
1997O . . . . . . .
1997P . . . . . . .
1997Q . . . . . . .
1997R . . . . . . .
1997S . . . . . . . .
1997ac . . . . . . .
1997af . . . . . . .
1997ai . . . . . . .
1997aj . . . . . . .
1997am . . . . . .
1997ap . . . . . .
0.458
0.354
0.425
0.374
0.420
0.372
0.378
0.453
0.465
0.498
0.655
0.400
0.615
0.480
0.450
0.388
0.570
0.490
0.495
0.656
0.828
0.450
0.430
0.580
0.763
0.526
0.172
0.619
0.592
0.550
0.180
0.374
0.472
0.430
0.657
0.612
0.320
0.579
0.450
0.581
0.416
0.830
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.005
0.001
0.001
0.030
0.001
0.001
0.001
0.001
0.010
0.010
0.001
0.001
0.001
0.010
0.010
0.001
0.001
0.001
0.001
0.001
0.001
0.010
0.001
0.001
0.001
0.010
0.001
0.001
0.010
0.001
0.010
0.001
0.001
0.010
22.12
22.08
21.52
21.28
22.37
21.82
22.14
22.60
22.71
23.02
22.62
21.75
22.53
22.64
22.44
22.08
22.70
22.46
22.19
23.08
23.53
22.66
22.58
22.90
23.56
22.68
20.04
23.25
23.73
22.93
20.19
22.97
22.52
22.01
23.28
23.03
21.38
22.96
22.25
22.55
21.97
23.20
0.10
0.10
0.21
0.06
0.06
0.07
0.08
0.07
0.04
0.07
0.03
0.03
0.07
0.04
0.07
0.04
0.03
0.03
0.03
0.07
0.10
0.07
0.03
0.06
0.41
0.05
0.02
0.08
0.10
0.05
0.01
0.07
0.04
0.03
0.05
0.05
0.03
0.07
0.05
0.06
0.03
0.07
0.03
0.11
0.03
0.10
0.42
0.10
0.21
0.07
0.07
0.07
0.07
0.12
0.11
0.35
0.61
0.06
0.13
0.11
0.09
0.13
0.18
0.15
0.08
0.13
0.20
0.16
0.16
0.13
0.07
0.08
0.10
0.09
0.10
0.09
0.11
0.11
0.09
0.09
0.14
0.11
0.11
0.13
[0.72
[0.58
[0.68
[0.61
[0.68
[0.61
[0.62
[0.71
[0.71
[0.71
[0.66
[0.65
[0.67
[0.72
[0.71
[0.63
[0.68
[0.72
[0.71
[0.66
[1.22
[0.71
[0.69
[0.68
[1.13
[0.70
[0.33
[0.67
[0.67
[0.69
[0.34
[0.61
[0.72
[0.69
[0.66
[0.67
[0.55
[0.68
[0.71
[0.68
[0.67
[1.23
22.81
22.55
22.17
21.79
22.63
22.32
22.55
23.24
23.35
23.66
23.21
22.27
23.10
23.00
22.53
22.66
23.25
23.06
22.82
23.62
24.58
23.22
23.19
23.45
24.49
23.21
20.20
23.80
24.33
23.53
20.42
23.50
23.14
22.60
23.83
23.59
21.83
23.54
22.81
23.12
22.52
24.30
23.11
22.38
22.13
21.72
22.55
22.26
22.58
23.17
23.33
23.71
23.27
22.36
23.19
22.96
22.51
22.65
23.27
23.10
22.83
23.57
24.65
23.17
23.13
23.46
24.47
23.15
20.17
23.80
24.42
23.51
20.43
23.52
23.11
22.57
23.83
23.69
21.86
23.48
22.83
23.09
22.57
24.32
0.46
0.33
0.49
0.22
0.25
0.20
0.37
0.25
0.30
0.25
0.21
0.19
0.25
0.24
0.23
0.20
0.22
0.20
0.19
0.28
0.54
0.23
0.22
0.23
0.53
0.20
0.18
0.28
0.37
0.25
0.17
0.24
0.19
0.18
0.23
0.21
0.18
0.22
0.30
0.22
0.20
0.22
Notes
(11)
EÈH
EÈH
BÈL
EÈH
EÈH
EÈH
H
H
H
C, D, GÈL
C, D, GÈL
H
H
H
H
BÈL
H
H
NOTE.ÈCol. (1) : IAU Name assigned to SCP supernova.
Col. (2) : Geocentric redshift of supernova or host galaxy.
Col. (3) : Redshift uncertainty.
Col. (4) : Peak magnitude from light-curve Ðt in observed band corresponding to rest-frame B-band (i.e., mpeak 4 mpeak
X
R
or mpeak).
I (5) : Uncertainty in Ðt peak magnitude.
Col.
Col. (6) : Galactic extinction in observed band corresponding to rest-frame B-band (i.e., A 4 A or A ) ; an uncerX
R
I
tainty of 10% is assumed.
Col. (7) : Representative K-correction (at peak) from observed band to B-band (i.e., K 4 K or K ) ; an uncerBX
BR
BI
tainty of 2% is assumed.
Col. (8) : B-band peak magnitude.
Col. (9) : Stretch luminosityÈcorrected e†ective B-band peak magnitude : meff 4 mpeak ] a(s [ 1) [ K [ A .
X uncertainties dueBXto theX intrinsic
Col. (10) : Total uncertainty in corrected B-band peak magnitude. This Bincludes
luminosity dispersion of SNe Ia of 0.17 mag, 10% of the Galactic extinction correction, 0.01 mag for K-corrections, 300
km s~1 to account for peculiar velocities, in addition to propagated uncertainties from the light-curve Ðts.
Col. (11) : Fits from which given supernova was excluded.
unconstrained Ðts have generally been used that can (and
do) lead to conÐdence regions that include the part of
parameter space with negative values for ) . The di†erences between the conÐdence regions thatM result from
Bayesian and classical analyses are small. We present the
Bayesian conÐdence regions in the Ðgures, since they are
somewhat more conservative ; i.e., they have larger conÐdence regions in the vicinity of particular interest near
" \ 0.
The residual dispersion in SN Ia peak magnitude after
correcting for the width-luminosity relation is small, about
0.17 mag, before applying any color-correction. This was
No. 2, 1999
) AND " FROM 42 HIGH-REDSHIFT SUPERNOVAE
571
TABLE 2
CALaN/TOLOLO SNE IA DATA
SN
(1)
z
(2)
p
z
(3)
mpeak
obs
(4)
ppeak
obs
(5)
A
B
(6)
K
BB
(7)
mpeak
B
(8)
mcorr
B
(9)
p corr
mB
(10)
1990O . . . . . .
1990af . . . . . .
1992P . . . . . . .
1992ae . . . . . .
1992ag . . . . . .
1992al . . . . . .
1992aq . . . . . .
1992bc . . . . . .
1992bg . . . . . .
1992bh . . . . . .
1992bl . . . . . .
1992bo . . . . . .
1992bp . . . . . .
1992br . . . . . .
1992bs . . . . . .
1993B . . . . . . .
1993O . . . . . .
1993ag . . . . . .
0.030
0.050
0.026
0.075
0.026
0.014
0.101
0.020
0.036
0.045
0.043
0.018
0.079
0.088
0.063
0.071
0.052
0.050
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
16.62
17.92
16.13
18.61
16.59
14.60
19.29
15.20
17.41
17.67
17.31
15.85
18.55
19.71
18.36
18.68
17.83
18.29
0.03
0.01
0.03
0.12
0.04
0.01
0.12
0.01
0.07
0.04
0.07
0.02
0.02
0.07
0.05
0.08
0.01
0.02
0.39
0.16
0.12
0.15
0.38
0.13
0.05
0.07
0.77
0.10
0.04
0.11
0.21
0.12
0.09
0.31
0.25
0.56
[0.00
]0.01
[0.01
]0.03
[0.01
[0.01
]0.05
[0.01
]0.00
]0.01
]0.01
[0.01
]0.04
]0.04
]0.03
]0.03
]0.01
]0.01
16.23
17.75
16.02
18.43
16.22
14.48
19.19
15.13
16.63
17.56
17.26
15.75
18.30
19.54
18.24
18.34
17.58
17.71
16.26
17.63
16.08
18.43
16.28
14.47
19.16
15.18
16.66
17.61
17.19
15.61
18.27
19.28
18.24
18.33
17.54
17.69
0.20
0.18
0.24
0.20
0.20
0.23
0.23
0.20
0.21
0.19
0.18
0.21
0.18
0.18
0.18
0.20
0.18
0.20
Notes
(11)
BÈL
BÈL
NOTE.ÈCol. (1) : IAU name assigned to Calan/Tololo supernova.
Col. (2) : Redshift of supernova or host galaxy in Local Group rest-frame.
Col. (3) : Redshift uncertainty.
Col. (4) : Peak magnitude from light-curve Ðt in observed B-band. Note that the template light curve used in the
Ðt is not identical to the template light curve used by Hamuy et al. (1995), so the best-Ðt peak magnitude may di†er
slightly.
Col. (5) : Uncertainty in Ðt peak magnitude.
Col. (6) : Galactic extinction in observed B-band ; an uncertainty of 10% is assumed.
Col. (7) : Representative K-correction from observed B-band to rest-frame B-band ; an uncertainty of 2% is
assumed.
Col. (8) : B-band peak magnitude.
Col. (9) : Stretch-luminosity corrected B-band peak magnitude.
Col. (10) : Total uncertainty in corrected B-band peak magnitude. This includes uncertainties due to the intrinsic
luminosity dispersion of SNe Ia of 0.17 mag, 10% of the Galactic extinction correction, 0.01 mag for K-corrections,
300 km s~1 to account for peculiar velocities, in addition to propagated uncertainties from the light-curve Ðts.
Col. (11) : Fits from which given supernova was excluded.
reported in Hamuy et al. (1996) for the low-redshift Calan/
Tololo supernovae, and it is striking that the same residual
is most consistent with the current 42 high-redshift supernovae (see ° 5). It is not clear from the current data sets,
however, whether this dispersion is best modeled as a
normal distribution (a Gaussian in Ñux space) or a lognormal distribution (a Gaussian in magnitude space). We
have therefore performed the Ðts in two ways : minimizing
s2 measured using either magnitude residuals or Ñux
residuals. The results are generally in excellent agreement,
but since the magnitude Ðts yield slightly larger conÐdence
regions, we have again chosen this more conservative alternative to report in this paper.
We have analyzed the total set of 60 low- plus highredshift supernovae in several ways, with the results of each
Ðt presented as a row of Table 3. The most inclusive
analyses are presented in the Ðrst two rows : Fit A is a Ðt to
the entire data set, while Ðt B excludes two supernovae that
are the most signiÐcant outliers from the average lightcurve width, s \ 1, and two of the remaining supernovae
that are the largest residuals from Ðt A. Figure 4 shows that
the remaining low- and high-redshift supernovae are well
matched in their light-curve width (the error-weighted
means are SsT
\ 0.99 ^ 0.01 and SsT \ 1.00
Hamuyresults robust with respect
SCP to the
^ 0.01) making the
width-luminosityÈrelation correction (see ° 4.5). Our
primary analysis, Ðt C, further excludes two supernovae
that are likely to be reddened, and is discussed in the following section.
Fits A and B give very similar results. Removing the two
large-residual supernovae from Ðt A yields indistinguishable results, while Figure 5a shows that the 68% and 90%
joint conÐdence regions for ) and ) still change very
M supernovae
"
little after also removing the two
with outlier
light-curve widths. The best-Ðt mass-density in a Ñat universe for Ðt A is, within a fraction of the uncertainty, the
same value as for Ðt B, )flat \ 0.26`0.09 (see Table 3). The
M Ðts is ~0.08
main di†erence between the
the goodness-of-Ðt : the
larger s2 per degree of freedom for Ðt A, s2 \ 1.76, indicates
l Ðt are probably
that the outlier supernovae included in this
not part of a Gaussian distribution and thus will not be
appropriately weighted in a s2 Ðt. The s2 per degree of
freedom for Ðt B, s2 \ 1.16, is over 300 times more probable
l indicates that the remaining 56 superthan that of Ðt A and
novae are a reasonable Ðt to the model, with no large statistical errors remaining unaccounted for.
Of the two large-residual supernovae excluded from the
Ðts after Ðt A, one is fainter than the best-Ðt prediction and
one is brighter. The photometric color excess (see ° 4.1) for
the fainter supernova, SN 1997O, has an uncertainty that is
too large to determine conclusively whether it is reddened.
The brighter supernova, SN 1994H, is one of the Ðrst seven
high-redshift supernovae originally analyzed in P97 and is
one of the few supernovae without a spectrum to conÐrm its
572
PERLMUTTER ET AL.
8
6
NSNe
8
Calan/Tololo
Survey
(a)
6
4
NSNe
2
Calan/Tololo
Survey
(a)
4
2
0
-2
-1
0
1
0
0.0
2
magnitude residual
0.5
1.0
1.5
2.0
stretch factor, s
14
12
Vol. 517
15
Supernova
Cosmology
Project
(b)
10
Supernova
Cosmology
Project
(b)
10
8
NSNe
NSNe
6
5
4
2
0
-2
-1
0
1
2
magnitude residual
FIG. 3.ÈDistribution of rest-frame B-band magnitude residuals from
the best-Ðt Ñat cosmology for the Ðt C supernova subset, for (a) 18
Calan/Tololo supernovae, at redshifts z ¹ 0.1 and (b) 42 supernovae from
the Supernova Cosmology Project, at redshifts between 0.18 and 0.83. The
darker shading indicates those residuals with uncertainties less than 0.35
mag, unshaded boxes indicate uncertainties greater than 0.35 mag, and
dashed boxes indicate the supernovae that are excluded from Ðt C. The
curves show the expected magnitude residual distributions if they are
drawn from normal distributions given the measurement uncertainties and
0.17 mag of intrinsic SN Ia dispersion. The low-redshift expected distribution matches a Gaussian with p \ 0.20 mag (with error on the mean of 0.05
mag), while the high-redshift expected distribution matches a Gaussian
with p \ 0.22 mag (with error on the mean of 0.04 mag).
classiÐcation as a SN Ia. After reanalysis with additional
calibration data and improved K-corrections, it remains the
brightest outlier in the current sample, but it a†ects the Ðnal
cosmological Ðts much less as one of 42 supernovae, rather
than 1 of 5 supernovae in the primary P97 analysis.
4.
SYSTEMATIC UNCERTAINTIES AND CROSS-CHECKS
With our large sample of 42 high-redshift supernovae, it
is not only possible to obtain good statistical uncertainties
on the measured parameters but also to quantify several
possible sources of systematic uncertainties. As discussed in
P97, the primary approach is to examine subsets of our data
that will be a†ected to lesser extents by the systematic
uncertainty being considered. The high-redshift sample is
now large enough that these subsets each contain enough
supernovae to yield results of high statistical signiÐcance.
4.1. Extragalactic Extinction
4.1.1. Color-Excess Distributions
Although we have accounted for extinction due to our
Galaxy, it is still probable that some supernovae are
0
0.0
0.5
1.0
1.5
2.0
stretch factor, s
FIG. 4.ÈDistribution of light-curve widths for (a) 18 Calan/Tololo
supernovae at redshifts z ¹ 0.1 and (b) 42 supernovae from the Supernova
Cosmology Project at redshifts between 0.18 and 0.83. The light-curve
widths are characterized by the ““ stretch factor,ÏÏ s, that stretches or contracts the time axis of a template SN Ia light curve to best Ðt the observed
light curve for each supernova (see Perlmutter et al. 1995a, 1997e ; Kim et
al. 1999 ; Goldhaber et al. 1999). The template has been time-dilated by a
factor 1 ] z before Ðtting to the observed light curves to account for the
cosmological lengthening of the supernova timescale (Goldhaber et al.
1995 ; Leibundgut et al. 1996a). The shading indicates those measurements
of s with uncertainties less than 0.1, and the dashed lines indicate the two
supernovae that are removed from the Ðts after Ðt A. These two excluded
supernovae are the most signiÐcant deviations from s \ 1 (the highest
stretch supernova in panel b has an uncertainty of ^0.23 and hence is not
a signiÐcant outlier from s \ 1) ; the remaining low- and high-redshift distributions have almost exactly the same error-weighted means :
SsT
\ 0.99 ^ 0.01 and SsT \ 1.00 ^ 0.01.
Hamuy
SCP
dimmed by host galaxy dust or intergalactic dust. For a
standard dust extinction law (Cardelli et al. 1989) the color,
B[V , of a supernova will become redder as the amount of
extinction, A , increases. We thus can look for any extincB between the low- and high-redshift supertion di†erences
novae by comparing their rest-frame colors. Since there is a
small dependence of intrinsic color on the light-curve width,
supernova colors can only be compared for the same stretch
factor ; for a more convenient analysis, we subtract out the
intrinsic colors so that the remaining color excesses can be
compared simultaneously for all stretch factors. To determine the rest-frame color excess E(B[V ) for each supernova, we Ðtted the rest-frame B and V photometry to the B
and V SN Ia light-curve templates, with one of the Ðtting
parameters representing the magnitude di†erence between
the two bands at their respective peaks. Note that these
light-curve peaks are D2 days apart, so the resulting
B [V
color parameter, which is frequently used to
max
max
No. 2, 1999
) AND " FROM 42 HIGH-REDSHIFT SUPERNOVAE
573
TABLE 3
FIT RESULTS
N
s2
DOF
)flat
M
P() [ 0)
"
Best Fit
() , ) )
M "
Inclusive Fits :
A . . . . . . 60
B . . . . . . 56
98
60
56
52
0.29`0.09
~0.08
0.26`0.09
~0.08
0.9984
0.9992
0.83, 1.42
0.85, 1.54
All supernovae
Fit A, but excluding two residual outliers and two stretch outliers
Primary Ðt :
C . . . . . . 54
56
50
0.28`0.09
~0.08
0.9979
0.73, 1.32
Fit B, but also excluding two likely reddened
Fit
Fit Description
Comparison Analysis Techniques :
D . . . . . . 54 53
51
0.25`0.10
0.9972
0.76, 1.48 No stretch correctiona
~0.09
E . . . . . . 53 62
49
0.29`0.12
0.9894
0.35, 0.76 Bayesian one-sided extinction correctedb
~0.10
E†ect of Reddest Supernovae :
F . . . . . . 51 59
47
0.26`0.09
0.9991
0.85, 1.54 Fit B supernovae with colors measured
~0.08
G . . . . . . 49 56
45
0.28`0.09
0.9974
0.73, 1.32 Fit C supernovae with colors measured
~0.08
H . . . . . . 40 33
36
0.31`0.11
0.9857
0.16, 0.50 Fit G, but excluding seven next reddest and two next faintest high-redshift supernovae
~0.09
Systematic Uncertainty Limits :
I . . . . . . . 54 56
50
0.24`0.09
0.9994
0.80, 1.52 Fit C with ]0.03 mag systematic o†set
~0.08
J . . . . . . . 54 57
50
0.33`0.10
0.9912
0.72, 1.20 Fit C with [0.04 mag systematic o†set
~0.09
Clumped Matter Metrics :
K . . . . . . 54 57
50
0.35`0.12
0.9984
2.90, 2.64 Empty beam metricc
~0.10
L . . . . . . 54 56
50
0.34`0.10
0.9974
0.94, 1.46 Partially Ðlled beam metric
~0.09
a A 0.24 mag intrinsic SNe Ia luminosity dispersion is assumed.
b Bayesian method of RPK96 with conservative prior (see text and Appendix) and 0.10 mag intrinsic SNe Ia luminosity dispersion.
c Assumes additional Bayesian prior of ) \ 3, ) \ 3.
M
"
describe supernova colors, is not a color measurement on a
particular day. The di†erence of this color parameter from
the B [ V
found for a sample of low-redshift supernovaemax
for themaxsame light-curve stretch-factor (Tripp 1998 ;
Kim et al. 1999 ; M. M. Phillips 1998, private
communication) does yield the rest-frame E(B[V ) color
excess for the Ðtted supernova.
For the high-redshift supernovae at 0.3 \ z \ 0.7, the
matching R- and I-band measurements take the place of the
rest-frame B and V measurements, and the Ðt B and V
light-curve templates are K-corrected from the appropriate
matching Ðlters, e.g., R(t) \ B(t) ] K (t) (Kim et al. 1996 ;
BR
Nugent et al. 1998). For the three supernovae
at z [ 0.75,
the observed R[I corresponds more closely to a rest-frame
U[B color than to a B[V color, so E(B[V ) is calculated
from rest-frame E(U[B) using the extinction law of Cardelli et al. (1989). Similarly, for the two SNe Ia at z D 0.18,
E(B[V ) is calculated from rest-frame E(V [R).
Figure 6 shows the color excess distributions for both the
low- and high-redshift supernovae after removing the color
excess due to our Galaxy. Six high-redshift supernovae are
not shown on this E(B[V ) plot, because six of the Ðrst
seven high-redshift supernovae discovered were not
observed in both R and I bands. The color of one lowredshift supernova, SN 1992bc, is poorly determined by the
V -band template Ðt and has also been excluded. Two supernovae in the high-redshift sample are [3 p red-and-faint
outliers from the mean in the joint probability distribution
of E(B[V ) color excess and magnitude residual from Ðt B.
These two, SNe 1996cg and 1996cn (Fig. 6 ; light shading),
are very likely reddened supernovae. To obtain a more
robust Ðt of the cosmological parameters, in Ðt C we
remove these supernovae from the sample. As can be seen
from the Ðt-C 68% conÐdence region of Figure 5a, these
likely reddened supernovae do not signiÐcantly a†ect any of
our results. The main distribution of 38 high-redshift supernovae thus is barely a†ected by a few reddened events. We
Ðnd identical results if we exclude the six supernovae
without color measurements (Ðt G in Table 3). We take Ðt C
to be our primary analysis for this paper, and in Figure 7 we
show a more extensive range of conÐdence regions for this
Ðt.
4.1.2. Cross-Checks on Extinction
The color-excess distributions of the Ðt C data set (with
the most signiÐcant measurements highlighted by dark
shading in Fig. 6) show no signiÐcant di†erence between the
low- and high-redshift means. The dashed curve drawn over
the high-redshift distribution of Figure 6 shows the
expected distribution if the low-redshift distribution had the
measurement uncertainties of the high-redshift supernovae
indicated by the dark shading. This shows that the
reddening distribution for the high-redshift supernovae is
consistent with the reddening distribution for the lowredshift supernovae, within the measurement uncertainties.
The error-weighted means of the low- and high-redshift distributions are almost identical : SE(B[V )T
\ 0.033
Hamuymag. We
^ 0.014 mag and SE(B[V )T
\ 0.035 ^ 0.022
SCP between the color excess
also Ðnd no signiÐcant correlation
and the statistical weight or redshift of the supernovae
within these two redshift ranges.
To test the e†ect of any remaining high-redshift
reddening on the Ðt C measurement of the cosmological
parameters, we have constructed a Ðt H subset of the highredshift supernovae that is intentionally biased to be bluer
than the low-redshift sample. We exclude the errorweighted reddest 25% of the high-redshift supernovae ; this
excludes nine high-redshift supernovae with the highest
error-weighted E(B[V ). We further exclude two supernovae that have large uncertainties in E(B[V ) but are signiÐcantly faint in their residual from Ðt C. This is a
somewhat conservative cut, since it removes the faintest of
the high-redshift supernovae, but it does ensure that the
error-weighted E(B[V ) mean of the remaining supernova
FIG. 5.ÈComparison of best-Ðt conÐdence regions in the ) -) plane. Each panel shows the result of Ðt C (shaded regions) compared with Ðts to di†erent
M supernovae,
"
subsets of supernovae, or variant analyses for the same subset of
to test the robustness of the Ðt C result. Unless otherwise indicated, the 68% and
90% conÐdence regions in the ) -) plane are shown after integrating the four-dimensional Ðts over the other two variables, M and a. The ““ noÈbig-bang ÏÏ
M
"
shaded region at the upper left, the Ñat-universe diagonal line, and the inÐnite expansion line are shown as in Fig. 7 for ease ofBcomparison. The panels are
described as follows : (a) Fit A of all 60 supernovae and Ðt B of 56 supernovae, excluding the two outliers from the light-curve width distribution and the two
remaining statistical outliers. Fit C further excludes the two likely reddened high-redshift supernovae. (b) Fit D of the same 54-supernova subset as in Ðt C,
but with no correction for the light-curve width-luminosity relation. (c) Fit H of the subset of supernovae with color measurements, after excluding the
reddest 25% (nine high-redshift supernovae) and the two faint high-redshift supernovae with large uncertainties in their color measurements. The close match
to the conÐdence regions of Ðt C indicates that any extinction of these supernovae is quite small and not signiÐcant in the Ðts of the cosmological parameters.
(d) 68% conÐdence region for Fit E of the 53 supernovae with color measurements from the Ðt B data set, but following the Bayesian reddening-correction
method of RPK96. This method, when used with any reasonably conservative prior (i.e., somewhat broader than the likely true extinction distribution ; see
text), can produce a result that is biased, with an approximate bias direction and worst-case amount indicated by the arrows. (e) Fits I and J. These are
identical to Ðt C but with 0.03 or 0.04 mag added or subtracted from each of the high-redshift supernova measurements to account for the full range of
identiÐed systematic uncertainty in each direction. Other hypothetical sources of systematic uncertainty (see Table 4B) are not included. ( f ) Fit M. This is a
separate two-parameter () and ) ) Ðt of just the high-redshift supernovae, using the values of M and a found from the low-redshift supernovae. The
M
B
dashed conÐdence regions show
the "approximate range of uncertainty from these two low-redshift-derived
parameters added to the systematic errors of Ðt J.
Future well-observed low-redshift supernovae can constrain this dashed-line range of uncertainty.
) AND " FROM 42 HIGH-REDSHIFT SUPERNOVAE
12
10
Calan/Tololo
Survey
(a)
8
NSNe
6
4
2
0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
E(B-V)
8
Supernova
Cosmology
Project
(b)
6
NSNe
4
2
0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
E(B-V)
FIG. 6.È(a) Rest-frame B[V color excess distribution for 17 of 18
Calan/Tololo supernovae (see text), corrected for Galactic extinction using
values from Schlegel et al. (1998). (b) Rest-frame B[V color excess for the
36 high-redshift supernovae for which rest-frame B[V colors were measured, also corrected for Galactic extinction. The darker shading indicates
those E(B[V ) measurements with uncertainties less than 0.3 mag,
unshaded boxes indicate uncertainties greater than 0.3 mag, and the light
shading indicates the two supernovae that are likely to be reddened based
on their joint probability in color excess and magnitude residual from Ðt B.
The dashed curve shows the expected high-redshift E(B[V ) distribution if
the low-redshift distribution had the measurement uncertainties of the
high-redshift supernovae indicated by the dark shading. Note that most of
the color-excess dispersion for the high-redshift supernovae is due to the
rest-frame V -band measurement uncertainties, since the rest-frame B-band
uncertainties are usually smaller.
575
magnitudes) as a D1 p upper bound on the systematic
uncertainty due to extinction by dust that reddens.
Note that the modes of both distributions appear to be at
zero reddening, and similarly the medians of the distributions are quite close to zero reddening : SE(B[V )Tmedian \
Hamuy
0.01 mag and SE(B[V )Tmedian \ 0.00 mag. This should be
SCP
taken as suggestive rather than conclusive, since the zero
point of the relationship between true color and stretch is
not tightly constrained by the current low-redshift SN Ia
data set. This apparent strong clustering of SNe Ia about
zero reddening has been noted in the past for low-redshift
supernova samples. Proposed explanations have been given
based on the relative spatial distributions of the SNe Ia and
the dust : Modeling by Hatano, Branch, & Deaton (1998) of
the expected extinction of SN Ia disk and bulge populations
viewed at random orientations shows an extinction distribution with a strong spiked peak near zero extinction
along with a broad, lower probability wing to higher extinction. This wing will be further suppressed by the observational selection against more reddened supernovae, since
they are dimmer. (For a Ñux-limited survey this suppression
factor is 10~aR*RB E(B~V)~a(s~1)+ B 10~1.6E(B~V), where a is
R
the slope of the supernova number counts.) We also note
that the high-redshift supernovae for which we have accurate measurements of apparent separation between SN and
host position (generally, those with HST imaging) appear to
be relatively far from the host center, despite our high
search sensitivity to supernovae in front of the host galaxy
core (see Pain et al. 1996 for search efficiency studies ; also
cf. Wang, HoŽÑich, & Wheeler 1997). If generally true for the
entire sample, this would be consistent with little extinction.
Our results, however, do not depend on the low- and
high-redshift color-excess distributions being consistent
with zero reddening. It is only important that the reddening
distributions for the low-redshift and high-redshift data sets
are statistically the same and that there is no correlation
between reddening and statistical weight in the Ðt of the
cosmological parameters. With both of these conditions
satisÐed, we Ðnd that our measurement of the cosmological
parameters is una†ected (to within the statistical error) by
any small remaining extinction among the supernovae in
the two data sets.
4.1.3. Analysis with Reddening Correction of Individual Supernovae
subset is a good indicator of any reddening that could a†ect
the cosmological parameters. The probability that the highredshift subset of Ðt H is redder in the mean than the lowredshift supernovae is less than 5% ; This subset is thus very
unlikely to be biased to fainter magnitudes by high-redshift
reddening. Even with nonstandard, ““ grayer ÏÏ dust that does
not cause as much reddening for the same amount of extinction, a conservative estimate of the probability that the
high-redshift subset of Ðt H is redder in the mean than the
low-redshift supernovae is still less than D17% for any
high-redshift value of R 4 A /E(B[V ) less than twice the
B
Bsame conÐdence levels are
low-redshift value. [These
obtained whether using Gaussian statistics, assuming a
normal distribution of E(B[V ) measurements, or using
bootstrap resampling statistics, based on the observed distribution.] The conÐdence regions of Figure 5c and the )flat
M
results in Table 3 show that the cosmological parameters
found for Ðt H di†er by less than half of a standard deviation from those for Ðt C. We take the di†erence of these
Ðts, 0.03 in )flat (which corresponds to less than 0.025 in
M
We have also performed Ðts using rest-frame B-band
magnitudes individually corrected for host galaxy extinction using A \ R E(B[V ) (implicitly assuming that the
B
extragalactic Bextinction
is all at the redshift of the host
galaxy). As a direct comparison between the treatment of
host galaxy extinction described above and an alternative
Bayesian method (RPK96), we applied it to the 53 SNe Ia
with color measurements in our Ðt C data set. We Ðnd that
our cosmological parameter results are robust with respect
to this change, although this method can introduce a bias
into the extinction corrections and hence the cosmological
parameters. In brief, in this method the Gaussian extinction
probability distribution implied by the measured colorexcess and its error is multiplied by an assumed a priori
probability distribution (the Bayesian prior) for the intrinsic
distribution of host extinctions. The most probable value of
the resulting renormalized probability distribution is taken
as the extinction, and following A. Riess (1998, private
communication) the second-moment is taken as the uncertainty. For this analysis, we choose a conservative prior (as
given in RPK96) that does not assume that the supernovae
576
PERLMUTTER ET AL.
are unextinguished but rather is somewhat broader than the
true extinction distribution where the majority of the previously observed supernovae apparently su†er very little
reddening. (If one alternatively assumes that the current
dataÏs extinction distribution is quite as narrow as that of
previously observed supernovae, one can choose a less conservative but more realistic narrow prior probability distribution, such as that of Hatano et al. 1998. This turns out
to be quite similar to our previous analysis in ° 4.1.1, since a
distribution like that of Hatano et al. 1998 has zero extinction for most supernovae.)
This Bayesian method with a conservative prior will only
brighten supernovae, never make them fainter, since it only
a†ects the supernovae with redder measurements than the
zero-extinction E(B[V ) value, leaving unchanged those
measured to be bluer than this. The resulting slight di†erence between the assumed and true reddening distributions
would make no di†erence in the cosmology measurements
if its size were the same at low and high redshifts. However,
since the uncertainties, phighvz , in the high-redshift data set
E(B~V)
E(B[V ) measurements are
larger on average than those of
the low-redshift data set, plowvz , this method can overE(B~V) on average relative to
correct the high-redshift supernovae
the low-redshift supernovae. Fortunately, as shown in the
Appendix, even an extreme case with a true distribution all
at zero extinction and a conservative prior would introduce
a bias in extinction A only of order 0.1 mag at worst for
our current low- andB high-redshift measurement uncertainties. The results of Ðt E are shown in Table 3 and as the
dashed contour in Figure 5d, where it can be seen that
compared to Ðt C this approach moves the best-Ðt value
much less than this and in the direction expected for this
e†ect (Fig. 5d ; arrows). The fact that )flat changes so little
M gives further confrom case C, even with the possible bias,
Ðdence in the cosmological results.
We can eliminate any such small bias of this method by
assuming no Bayesian prior on the host-galaxy extinction,
allowing extinction corrections to be negative in the case of
supernovae measured to be bluer than the zero-extinction
E(B[V ) value. As expected, we recover the unbiased results
within error but with larger uncertainties, since the Bayesian prior also narrows the error bars in the method of
RPK96. However, there remains a potential source of bias
when correcting for reddening : the e†ective ratio of total to
selective extinction, R , could vary for several reasons.
B be due to host galaxy dust at the
First, the extinction could
supernovaÏs redshift or intergalactic dust at lower redshifts,
where it will redden the supernova less, since it is acting on
a redshifted spectrum. Second, R may be sensitive to dust
B in the dust extinction
density, as indicated by variations
laws between various sight lines in the Galaxy (Clayton &
Cardelli 1988 ; Gordon & Clayton 1998). Changes in metallicity might be expected to be a third possible cause of R
B
evolution, since metallicity is one dust-related quantity
known to evolve with redshift (Pettini et al. 1997), but fortunately it appears not to signiÐcantly alter R , as evidenced
by the similarity of the optical portions ofBthe extinction
curves of the Galaxy, the LMC, and the SMC (Pei 1992 ;
Gordon & Clayton 1998). Three-Ðlter photometry of highredshift supernovae currently in progress with the HST will
help test for such di†erences in R .
To avoid these sources of bias,B we consider it important
to use and compare both analysis approaches : the rejection
of reddened supernovae and the correction of reddened
supernovae. We do Ðnd consistency in the results calculated
Vol. 517
both ways. The advantages of the analyses with reddening
corrections applied to individual supernovae (with or
without a Bayesian prior on host-galaxy extinction) are outweighed by the disadvantages for our sample of highredshift supernovae ; although, in principle, by applying
reddening corrections the intrinsic magnitude dispersion of
SNe Ia can be reduced from an observed dispersion of 0.17
mag to approximately 0.12 mag, in practice the net
improvement for our sample is not signiÐcant, since uncertainties in the color measurements often dominate. We have
therefore chosen for our primary analysis to follow the Ðrst
procedure discussed above, removing the likely reddened
supernovae (Ðt C) and then comparing color-excess means.
The systematic di†erence for Ðt H, which rejects the reddest
and the faintest high-redshift supernovae, is already quite
small, and we avoid introducing additional actual and possible biases. Of course, neither approach avoids biases if R
B
at high redshift is so large [[2R (z \ 0)] that dust does not
B
redden the supernovae enough to be distinguished and this
dust makes more than a few supernovae faint.
4.2. Malmquist Bias and Other L uminosity Biases
In the Ðt of the cosmological parameters to the
magnitude-redshift relation, the low-redshift supernova
magnitudes primarily determine M and the widthB
luminosity slope a, and then the comparison
with the highredshift supernova magnitudes primarily determines )
M
and ) . Both low- and high-redshift supernova samples can
"
be biased toward selecting the brighter tail of any distribution in supernova detection magnitude for supernovae
found near the detection threshold of the search (classical
Malmquist bias ; Malmquist 1924, 1936). A widthluminosity relation Ðt to such a biased population would
have a slope that is slightly too shallow and a zero point
slightly too bright. A second bias is also acting on the supernova samples, selecting against supernovae on the narrowÈ
light-curve side of the width-luminosity relation, since such
supernovae are detectable for a shorter period of time. Since
this bias removes the narrowest/faintest supernova light
curves preferentially, it culls out the part of the widthbrightness distribution most subject to Malmquist bias and
moves the resulting best-Ðt slope and zero point closer to
their correct values.
If the Malmquist bias is the same in both data sets, then it
is completely absorbed by M , and a and does not a†ect the
B
cosmological parameters. Thus
our principal concern is
that there could be a di†erence in the amount of bias
between the low- and high-redshift samples. Note that
e†ects peculiar to photographic supernovae searches, such
as saturation in galaxy cores, that might in principle select
slightly di†erent SNe Ia subpopulations should not be
important in determining luminosity bias, because lightcurve stretch compensates for any such di†erences. Moreover, Figure 4 shows that the high-redshift SNe Ia we have
discovered have a stretch distribution entirely consistent
with those discovered in the Calan/Tololo search.
To estimate the Malmquist bias of the high-redshiftsupernova sample, we Ðrst determined the completeness of
our high-redshift searches as a function of magnitude,
through an extensive series of tests inserting artiÐcial supernovae into our images (see Pain et al. 1996). We Ðnd that
roughly 30% of our high-redshift supernovae were detected
within twice the SN Ia intrinsic luminosity dispersion of the
50% completeness limit, where the above biases might be
important. This is consistent with a simple model where the
No. 2, 1999
) AND " FROM 42 HIGH-REDSHIFT SUPERNOVAE
supernova number counts follow a power-law slope of 0.4
mag~1, similar to that seen for comparably distant galaxies
(Smail et al. 1995). For a Ñux-limited survey of standard
candles having the light-curve width-corrected luminosity
dispersion for SNe Ia of D0.17 mag and this number-count
power-law slope, we can calculate that the classical Malmquist bias should be 0.03 mag (see, e.g., Mihalas & Binney
1981 for a derivation of the classical Malmquist bias). (Note
that this estimate is much smaller than the Malmquist bias
a†ecting other cosmological distance indicators because of
the much smaller intrinsic luminosity dispersion of SNe Ia.)
These high-redshift supernovae, however, are typically
detected before maximum, and their detection magnitudes
and peak magnitudes have a correlation coefficient of only
0.35, so the e†ects of classical Malmquist bias should be
diluted. Applying the formalism of Willick (1994) we estimate that the decorrelation between detection magnitude
and peak magnitude reduces the classical Malmquist bias in
the high-redshift sample to only 0.01 mag. The redshift and
stretch distributions of the high-redshift supernovae that
are near the 50% completeness limit track those of the
overall high-redshift sample, again suggesting that Malmquist biases are small for our data set.
We cannot make an exactly parallel estimate of Malmquist bias for the low-redshiftÈsupernova sample, because
we do not have information for the Calan/Tololo data set
concerning the number of supernovae found near the detection limit. However, the amount of classical Malmquist bias
should be similar for the Calan/Tololo supernovae, since
the amount of bias is dominated by the intrinsic luminosity
dispersion of SNe Ia, which we Ðnd to be the same for the
low-redshift and high-redshift samples (see ° 5). Figure 4
shows that the stretch distributions for the high- and lowredshift samples are very similar, so that the compensating
e†ects of stretch bias should also be similar in the two data
sets. The major source of di†erence in the bias is expected to
be due to the close correlation between the detection magnitude and the peak magnitude for the low-redshift supernova
search, since this search tended not to Ðnd the supernovae
as early before peak as the high-redshift search. In addition,
the number-counts at low-redshift should be somewhat
steeper (Maddox et al. 1990). We thus expect the
Calan/Tololo supernovae to have a bias closer to that
obtained by direct application of the classical Malmquist
bias formula, 0.04 mag. One might also expect
““ inhomogeneous Malmquist bias ÏÏ to be more important
for the low-redshift supernovae, since in smaller volumes of
space inhomogeneities in the host galaxy distribution might
by chance put more supernovae near the detection limit
than would be expected for a homogeneous distribution.
However, after averaging over all the Calan/Tololo
supernova-search Ðelds, the total low-redshift volume
searched is large enough that we expect galaxy count Ñuctuations of only D4%, so the classical Malmquist bias is
still a good approximation.
We believe that both these low- and high-redshift biases
may be smaller and even closer to each other, because of the
mitigating e†ect of the bias against detection of low-stretch
supernovae, discussed above. However, to be conservative
we take the classical Malmquist bias of 0.04 mag for the
low-redshift data set and the least biased value of 0.01 mag
for the high-redshift data set, and we consider systematic
uncertainty from this source to be the di†erence, 0.03 mag,
in the direction of low-redshift supernovae more biased
than high-redshift. In the other direction, i.e., for high-
577
redshift supernovae more biased than low-redshift, we consider the extreme case of a fortuitously unbiased
low-redshift sample and take the systematic uncertainty
bound to be the 0.01 mag bias of the high-redshift sample.
(In this direction any systematic error is less relevant to the
question of the existence of a cosmological constant.)
4.3. Gravitational L ensing
As discussed in P97, the clumping of mass in the universe
could leave the line-of-sight to most of the supernovae
underdense, while occasional supernovae may be seen
through overdense regions. The latter supernovae could be
signiÐcantly brightened by gravitational lensing, while the
former supernovae would appear somewhat fainter. With
enough supernovae, this e†ect will average out (for inclusive
Ðts such as Ðt A, which include outliers), but the most overdense lines of sight may be so rare that a set of 42 supernovae may only sample a slightly biased (fainter) set. The
probability distribution of these ampliÐcations and deampliÐcations has previously been studied both analytically
and by Monte Carlo simulations. Given the acceptance
window of our supernova search, we can integrate the probability distributions from these studies to estimate the bias
due to ampliÐed or deampliÐed supernovae that may be
rejected as outliers. This average (de)ampliÐcation bias is
less than 1% at the redshifts of our supernovae for simulations based on isothermal spheres the size of typical galaxies
(Holz & Wald 1998), N-body simulations using realistic
mass power spectra (Wambsganss, Cen, & Ostriker 1998),
and the analytic models of Frieman (1996).
It is also possible that the small-scale clumping of matter
is more extreme, for example, if signiÐcant amounts of mass
were in the form of compact objects such as MACHOs. This
could lead to many supernova sight lines that are not just
underdense but nearly empty. Once again, only the very
rare line of sight would have a compact object in it, amplifying the supernova signal. To Ðrst approximation, with 42
supernovae we would see only the nearly empty beams and
thus only deampliÐcations. The appropriate luminositydistance formula in this case is not the FriedmannRobertson-Walker (FRW) formula but rather the ““ partially
Ðlled beam ÏÏ formula with a mass Ðlling factor, g B 0 (see
Kantowski 1998 and references therein). We present the
results of the Ðt of our data (Ðt K) with this luminositydistance formula (as calculated using the code of Kayser,
Helbig, & Schramm 1996) in Figure 8. A more realistic limit
on this pointlike mass density can be estimated, because we
would expect such pointlike masses to collect into the gravitational potential wells already marked by galaxies and
clusters. Fukugita, Hogan, & Peebles (1998) estimate an
upper limit of ) \ 0.25 on the mass that is clustered like
M 8, we also show the conÐdence region
galaxies. In Figure
from Ðt L, assuming that only the mass density contribution
up to ) \ 0.25 is pointlike, with Ðlling factor g \ 0, and
M to 0.75 at ) \ 1. We see that at low mass
that g rises
M
density, the FRW Ðt is already
very close to the nearly
empty-beam (g B 0) scenario, so the results are quite
similar. At high mass density, the results diverge, although
only minimally for Ðt L ; the best Ðt in a Ñat universe is
)flat \ 0.34`0.10.
M
~0.09
4.4. Supernova Evolution and Progenitor
Environment Evolution
The spectrum of a SN Ia on any given point in its light
curve reÑects the complex physical state of the supernova
578
PERLMUTTER ET AL.
FIG. 7.ÈBest-Ðt conÐdence regions in the ) -) plane for our primary
M "
analysis, Ðt C. The 68%, 90%, 95%, and 99% statistical
conÐdence regions
in the ) È) plane are shown, after integrating the four-dimensional Ðt
M
"
over M and a. (See footnote 11 for a link to the table of this twoB
dimensional
probability distribution.) See Fig. 5e for limits on the small
shifts in these contours due to identiÐed systematic uncertainties. Note that
the spatial curvature of the universeÈopen, Ñat, or closedÈis not determinative of the future of the universeÏs expansion, indicated by the nearhorizontal solid line. In cosmologies above this near-horizontal line the
universe will expand forever, while below this line the expansion of the
universe will eventually come to a halt and recollapse. This line is not quite
horizontal, because at very high mass density there is a region where the
mass density can bring the expansion to a halt before the scale of the
universe is big enough that the mass density is dilute with respect to the
cosmological constant energy density. The upper-left shaded region,
labeled ““ no big bang,ÏÏ represents ““ bouncing universe ÏÏ cosmologies with
no big bang in the past (see Carroll et al. 1992). The lower right shaded
region corresponds to a universe that is younger than the oldest heavy
elements (Schramm 1990) for any value of H º 50 km s~1 Mpc~1.
0
on that day : the distribution, abundances, excitations, and
velocities of the elements that the photons encounter as they
leave the expanding photosphere all imprint on the spectra.
So far, the high-redshift supernovae that have been studied
have light-curve shapes just like those of low-redshift supernovae (see Goldhaber et al. 1999), and their spectra show
the same features on the same day of the light curve as their
low-redshift counterparts having comparable light-curve
width. This is true all the way out to the z \ 0.83 limit of the
current sample (Perlmutter et al. 1998b). We take this as a
strong indication that the physical parameters of the supernova explosions are not evolving signiÐcantly over this time
span.
Theoretically, evolutionary e†ects might be caused by
changes in progenitor populations or environments. For
Vol. 517
example, lower metallicity and more massive SN Iaprogenitor binary systems should be found in younger
stellar populations. For the redshifts that we are considering, z \ 0.85, the change in average progenitor masses may
be small (Ruiz-Lapuente, Canal, & Burkert 1997 ; RuizLapuente 1998). However, such progenitor mass di†erences
or di†erences in typical progenitor metallicity are expected
to lead to di†erences in the Ðnal C/O ratio in the exploding
white dwarf and hence a†ect the energetics of the explosion.
The primary concern here would be if this changed the
zero-point of the width-luminosity relation. We can look for
such changes by comparing light curve rise times between
low- and high-redshift supernova samples, since this is a
sensitive indicator of explosion energetics. Preliminary indications suggest that no signiÐcant rise-time change is seen,
with an upper limit of [1 day for our sample (see forthcoming high-redshift studies of Goldhaber et al. 1999 and
Nugent et al. 1998 and low-redshift bounds from Vacca &
Leibundgut 1996, Leibundgut et al. 1996b, and Marvin &
Perlmutter 1989). This tight a constraint on rise-time
change would theoretically limit the zero-point change to
less than D0.1 mag (see Nugent et al. 1995 ; HoŽÑich,
Wheeler, & Thielemann 1998).
A change in typical C/O ratio can also a†ect the ignition
density of the explosion and the propagation characteristics
of the burning front. Such changes would be expected to
appear as di†erences in light-curve timescales before and
after maximum (HoŽÑich & Khokhlov 1996). Preliminary
indications of consistency between such low- and highredshift light-curve timescales suggest that this is probably
not a major e†ect for our supernova samples (Goldhaber et
al. 1999).
Changes in typical progenitor metallicity should also
directly cause some di†erences in SN Ia spectral features
(HoŽÑich et al. 1998). Spectral di†erences big enough to
a†ect the B- and V -band light curves (see, e.g., the extreme
mixing models presented in Fig. 9 of HoŽÑich et al. 1998)
should be clearly visible for the best signal-to-noise ratio
spectra we have obtained for our distant supernovae, yet
they are not seen (Filippenko et al. 1998 ; Hook et al. 1998).
The consistency of slopes in the light-curve widthluminosity relation for the low- and high-redshift supernovae can also constrain the possibility of a strong
metallicity e†ect of the type that HoŽÑich et al. (1998)
describes.
An additional concern might be that even small changes
in spectral features with metallicity could in turn a†ect the
calculations of K-corrections and reddening corrections.
This e†ect, too, is very small, less than 0.01 mag, for photometric observations of SNe Ia conducted in the rest-frame B
or V bands (see Figs. 8 and 10 of HoŽÑich et al. 1998), as is
the case for almost all of our supernovae. (Only two of our
supernovae have primary observations that are sensitive to
the rest-frame U band, where the magnitude can change by
D0.05 mag, and these are the two supernovae with the
lowest weights in our Ðts, as shown by the error bars of Fig.
2. In general the I-band observations, which are mostly
sensitive to the rest-frame B band, provide the primary light
curve at redshifts above 0.7.)
The above analyses constrain only the e†ect of
progenitor-environment evolution on SN Ia intrinsic luminosity ; however, the extinction of the supernova light could
also be a†ected, if the amount or character of the dust
evolves, e.g., with host galaxy age. In ° 4.1, we limited the
No. 2, 1999
) AND " FROM 42 HIGH-REDSHIFT SUPERNOVAE
size of this extinction evolution for dust that reddens, but
evolution of ““ gray ÏÏ dust grains larger than D0.1 km, which
would cause more color-neutral optical extinction, can
evade these color measurements. The following two analysis
approaches can constrain both evolution e†ects, intrinsic
SN Ia luminosity evolution and extinction evolution. They
take advantage of the fact that galaxy properties such as
formation age, star formation history, and metallicity are
not monotonic functions of redshift, so even the lowredshift SNe Ia are found in galaxies with a wide range of
ages and metallicities. It is a shift in the distribution of relevant host-galaxy properties occurring between z D 0 and
z D 0.5 that could cause any evolutionary e†ects.
W idth-luminosity relation across low-redshift environments : To the extent that low-redshift SNe Ia arise from
progenitors with a range of metallicities and ages, the lightcurve width-luminosity relation discovered for these supernovae can already account for these e†ects (cf. Hamuy et al.
1995, 1996). When corrected for the width-luminosity relation, the peak magnitudes of low-redshift SNe Ia exhibit a
very narrow magnitude dispersion about the Hubble line,
with no evidence of a signiÐcant progenitor-environment
di†erence in the residuals from this Ðt. It therefore does not
matter if the population of progenitors evolves such that the
measured light-curve widths change, since the widthluminosity relation apparently is able to correct for these
changes. It will be important to continue to study further
nearby SNe Ia to test this conclusion with as wide a range
of host-galaxy ages and metallicities as possible.
Matching low- and high-redshift environments : Galaxies
with di†erent morphological classiÐcations result from different evolutionary histories. To the extent that galaxies
with similar classiÐcations have similar histories, we can
also check for evolutionary e†ects by using supernovae in
our cosmology measurements with matching host galaxy
classiÐcations. If the same cosmological results are found
for each measurement based on a subset of low- and highredshift supernovae sharing a given host-galaxy classiÐcation, we can rule out many evolutionary scenarios. In
the simplest such test, we compare the cosmological parameters measured from low- and high-redshift elliptical host
galaxies with those measured from low- and high-redshift
spiral host galaxies. Without high-resolution host-galaxy
images for most of our high-redshift sample, we currently
can only approximate this test for the smaller number of
supernovae for which the host-galaxy spectrum gives a
strong indication of galaxy classiÐcation. The resulting sets
of nine elliptical-host and eight spiral-host high-redshift
supernovae are matched to the four elliptical-host and 10
spiral-host low-redshift supernovae (based on the morphological classiÐcations listed in Hamuy et al. 1996 and
excluding two with SB0 hosts). We Ðnd no signiÐcant
change in the best-Ðt cosmology for the elliptical hostgalaxy subset (with both the low- and high-redshift subsets
about 1 p brighter than the mean of the full sets) and a small
(\1 p) shift lower in )flat for the spiral host-galaxy subset.
M of these subset results is encourAlthough the consistency
aging, the uncertainties are still large enough
(approximately twice the Ðt C uncertainties) that this test
will need to await the host-galaxy classiÐcation of the full
set of high-redshift supernovae and a larger low-redshift
supernova sample.
4.5. Further Cross-Checks
We have checked several other possible e†ects that might
579
bias our results by Ðtting di†erent supernova subsets and
using alternative analyses :
Sensitivity to width-luminosity correction : Although the
light-curve width correction provides some insurance
against supernova evolution biasing our results, Figure 4
shows that almost all of the Ðt C supernovae at both low
and high redshift are clustered tightly around the most
probable value of s \ 1, the standard width for a B-band
Leibundgut SN Ia template light curve. Our results are
therefore rather robust with respect to the actual choice of
width-luminosity relation. We have tested this sensitivity by
reÐtting the supernovae of Ðt C but with no widthluminosity correction. The results (Ðt D), as shown in
Figure 5b and listed in Table 3, are in extremely close agreement with those of the light-curveÈwidth-corrected Ðt C.
The statistical uncertainties are also quite close ; the lightcurve width correction does not signiÐcantly improve the
statistical dispersion for the magnitude residuals because of
the uncertainty in s, the measured light-curve width. It is
clear that the best-Ðt cosmology does not depend strongly
on the extra degree of freedom allowed by including the
width-luminosity relation in the Ðt.
Sensitivity to nonÈSN Ia contamination : We have tested
for the possibility of contamination by nonÈSN Ia events
masquerading as SNe Ia in our sample by performing a Ðt
after excluding any supernovae with less certain SN Ia spectroscopic and photometric identiÐcation. This selection
removes the large statistical outliers from the sample. In
part, this may be because the host-galaxy contamination
that can make it difficult to identify the supernova spectrum
can also increase the odds of extinction or other systematic
uncertainties in photometry. For this more ““ pure ÏÏ sample
of 43 supernovae, we Ðnd )flat \ 0.33`0.10, just over half of
~0.09
a standard deviation from ÐtMC.
Sensitivity to galactic extinction model : Finally, we have
tested the e†ect of the choice of Galactic extinction model,
with a Ðt using the model of Burstein & Heiles (1982), rather
than Schlegel et al. (1998). We Ðnd no signiÐcant di†erence
in the best-Ðt cosmological parameters, although we note
that the extinction near the Galactic pole is somewhat
larger in the Schlegel et al. model and that this leads to a
D0.03 mag larger average o†set between the low-redshift
supernova B-band observations and the high-redshift
supernovae R-band observations.
5.
RESULTS AND ERROR BUDGET
From Table 3 and Figure 5a, it is clear that the results of
Ðts A, B, and C are quite close to each other, so we can
conclude that our measurement is robust with respect to the
choice of these supernova subsets. The inclusive Ðts A and B
are the Ðts with the least subjective selection of the data.
They already indicate the main cosmological results from
this data set. However, to make our results robust with
respect to host-galaxy reddening, we use Ðt C as our
primary Ðt in this paper. For Ðt C, we Ðnd )flat \ 0.28`0.09
M a poor~0.08
in a Ñat universe. Cosmologies with ) \ 0 are
Ðt to
"
the data at the 99.8% conÐdence level. The contours of
Figure 7 more fully characterize the best-Ðt conÐdence
regions.12
The residual plots of Figures 2b and 2c indicate that the
best-Ðt )flat in a Ñat universe is consistent across the redshift rangeM of the high-redshift supernovae. Figure 2c shows
12 The data are available at http ://www-supernova.lbl.gov.
580
PERLMUTTER ET AL.
FIG. 8.ÈBest-Ðt 68% and 90% conÐdence regions in the ) -) plane
for cosmological models with small scale clumping of matter M(e.g.," in the
form of MACHOs) compared with the FRW model of Ðt C, with smooth
small-scale matter distribution. The shaded contours (Ðt C) are the conÐdence regions Ðt to a FRW magnitude-redshift relation. The extended
conÐdence strips (Ðt K) are for a Ðt of the Ðt C supernova set to an ““ empty
beam ÏÏ cosmology, using the ““ partially Ðlled beam ÏÏ magnitude-redshift
relation with a Ðlling factor g \ 0, representing an extreme case in which
all mass is in compact objects. The Ðt L unshaded contours represent a
somewhat more realistic partially ÐlledÈbeam Ðt, with clumped matter
(g \ 0) only accounting for up to ) \ 0.25 of the critical mass density
and any matter beyond that amountMsmoothly distributed (i.e., g rising to
0.75 at ) \ 1).
M
the residuals normalized by uncertainties ; their scatter can
be seen to be typical of a normal-distributed variable, with
the exception of the two outlier supernovae that are
removed from all Ðts after Ðt A, as discussed above. Figure 3
compares the magnitude-residual distributions (the projections of Fig. 2b) to the Gaussian distributions expected
given the measurement uncertainties and an intrinsic dispersion of 0.17 mag. Both the low- and high-redshift distributions are consistent with the expected distributions ;
the formal calculation of the SN Ia intrinsic-dispersion
component of the observed magnitude dispersion
(p2
\ p2
[ p2
) yields p
\ 0.154
measurement
intrinsic
^intrinsic
0.04 for observed
the low-redshift
distribution and
p
\
intrinsic
0.157 ^ 0.025 for the high-redshift distribution. The
s2 per
degree of freedom for this Ðt, s2 \ 1.12, also indicates that
l
the Ðt model is a reasonable description
of the data. The
narrow intrinsic dispersionÈwhich does not increase at
high redshiftÈprovides additional evidence against an
increase in extinction with redshift. Even if there is gray dust
that dims the supernovae without reddening them, the dispersion would increase, unless the dust is distributed very
uniformly.
A Ñat, ) \ 0 cosmology is a quite poor Ðt to the data.
The () , )" ) \ (1, 0) line on Figure 2b shows that 38 out of
M "
42 high-redshift
supernovae are fainter than predicted for
this model. These supernovae would have to be over 0.4
mag brighter than measured (or the low-redshift supernovae 0.4 mag fainter) for this model to Ðt the data.
The () , ) ) \ (0, 0) upper solid line on Figure 2a shows
M are
" still not a good Ðt to an ““ empty universe, ÏÏ
that the data
with zero mass density and cosmological constant. The
Vol. 517
high-redshift supernovae are as a group fainter than predicted for this cosmology ; in this case, these supernovae
would have to be almost 0.15 mag brighter for this empty
cosmology to Ðt the data, and the discrepancy is even larger
for ) [ 0. This is reÑected in the high probability (99.8%)
M
of ) [ 0.
"
As discussed in Goobar & Perlmutter (1995), the slope of
the contours in Figure 7 is a function of the supernova
redshift distribution ; since most of the supernovae reported
here are near z D 0.5, the conÐdence region is approximately Ðtted by 0.8) [ 0.6) B [0.2 ^ 0.1. (The
M
"
orthogonal linear combination, which is poorly constrained, is Ðtted by 0.6) ] 0.8) B 1.5 ^ 0.7.) In P97 we
M
"
emphasized that the well-constrained linear combination is
not parallel to any contour of constant currentdeceleration-parameter, q 4 ) /2 [ ) ; the accelerating/
0
M
"
decelerating universe line of Figure 9 shows one such
contour at q \ 0. Note that with almost all of the con0 above this line, only currently accelerating
Ðdence region
universes Ðt the data well. As more of our highest redshift
supernovae are analyzed, the long dimension of the conÐdence region will shorten.
5.1. Error Budget
Most of the sources of statistical error contribute a statistical uncertainty to each supernova individually and are
included in the uncertainties listed in Tables 1 and 2, with
small correlations between these uncertainties given in the
correlated-error matrices.13 These supernova-speciÐc statistical uncertainties include the measurement errors on SN
peak magnitude, light-curve stretch factor, and absolute
photometric calibration. The two sources of statistical error
that are common to all the supernovae are the intrinsic
dispersion of SN Ia luminosities after correcting for the
width-luminosity relation, taken as 0.17 mag, and the redshift uncertainty due to peculiar velocities, which are taken
as 300 km s~1. Note that the statistical error in M and a
B inteare derived quantities from our four-parameter Ðts. By
grating the four-dimensional probability distributions over
these two variables, their uncertainties are included in the
Ðnal statistical errors.
All uncertainties that are not included in the statistical
error budget are treated as systematic errors for the purposes of this paper. In °° 2 and 4, we have identiÐed and
bounded four potentially signiÐcant sources of systematic
uncertainty : (1) the extinction uncertainty for dust that
reddens, bounded at \0.025 mag, the maximal e†ect of the
nine reddest and two faintest of the high-redshift supernovae ; (2) the di†erence between the Malmquist bias of the
low- and high-redshift supernovae, bounded at ¹0.03 mag
for low-redshift supernovae biased intrinsically brighter
than high-redshift supernovae and at \0.01 mag for highredshift supernovae biased brighter than low-redshift supernovae ; (3) the cross-Ðlter K-correction uncertainty of \0.02
mag ; and (4) the \0.01 mag uncertainty in K-corrections
and reddening corrections due to the e†ect of progenitor
metallicity evolution on the rest-frame B-band spectral features. We take the total identiÐed systematic uncertainty to
be the quadrature sum of the sources : ]0.04 mag in the
direction of spuriously fainter high-redshift or brighter lowredshift supernovae and [0.03 mag in the opposite direction.
Note that we treat the possibility of gravitational lensing
13 The data are available at http ://www-supernova.lbl.gov.
No. 2, 1999
) AND " FROM 42 HIGH-REDSHIFT SUPERNOVAE
581
The measurement error of the cosmological parameters
has contributions from both the low- and high-redshift
supernova data sets. To identify the approximate relative
importance of these two contributory sources, we reanalyzed the Ðt C data set, Ðrst Ðtting M and a to the
B
low-redshift data set (this is relatively insensitive to cosmological model) and then Ðtting ) and ) to the highM
"
redshift data set. (This is only an approximation, since it
neglects the small inÑuence of the low-redshift supernovae
on ) and ) and of the high-redshift supernovae on M
M in the "standard four-parameter Ðt.) Figure 5 shows
B
and a,
this ) -) Ðtted as a solid contour (labeled Ðt M) with the
M "
1 p uncertainties on M and a included with the systematic
B
uncertainties in the dashed-line conÐdence contours. This
approach parallels the analyses of Permutter et al. (1997e,
1998b ; 1997f) and thus also provides a direct comparison
with the earlier results. We Ðnd that the more important
contribution to the uncertainty is currently due to the lowredshift supernova sample. If three times as many wellobserved low-redshift supernovae were discovered and
included in the analysis, then the statistical uncertainty
from the low-redshift data set would be smaller than the
other sources of uncertainty.
We summarize the relative statistical and systematic
uncertainty contributions in Table 4.
6.
FIG. 9.ÈIsochrones of constant H t , the age of the universe relative to
0 and 90% conÐdence regions in
the Hubble time, H~1, with the best-Ðt0 68%
0 the primary analysis, Ðt C. The isochrones are labeled
the ) -) plane for
M case
" of H \ 63 km s~1 Mpc~1, representing a typical value found
for the
0 Ia (Hamuy et al. 1996 ; RPK96 ; Saha et al. 1997 ; Tripp
from studies of SNe
1998). If H were taken to be 10% larger (i.e., closer to the values in
0 al. 1998), the age labels would be 10% smaller. The diagonal
Freedman et
line labeled accelerating/decelerating is drawn for q 4 ) /2 [ ) \ 0
0
"
and divides the cosmological models with an accelerating
orM decelerating
expansion at the present time.
due to small-scale clumping of mass as a separate analysis
case rather than as a contributing systematic error in our
primary analysis ; the total systematic uncertainty applies to
this analysis as well. There are also several more hypothetical sources of systematic error discussed in ° 4, which are
not included in our calculation of identiÐed systematics.
These include gray dust [with R (z \ 0.5) [ 2R (z \ 0)]
B that might change
B
and any SN Ia evolutionary e†ects
the
zero point of the light-curve width-luminosity relation. We
have presented bounds and tests for these e†ects, which give
preliminary indications that they are not large sources of
uncertainty, but at this time they remain difficult to quantify, at least partly because the proposed physical processes
and entities that might cause the e†ects are not completely
deÐned.
To characterize the e†ect of the identiÐed systematic
uncertainties, we have reÐt the supernovae of Ðt C for the
hypothetical case (Ðt J) in which each of the high-redshift
supernovae were discovered to be 0.04 mag brighter than
measured, or, equivalently, the low-redshift supernovae
were discovered to be 0.04 mag fainter than measured.
Figure 5e and Table 3 show the results of this Ðt. The bestÐt Ñat-universe )flat varies from that of Ðt C by 0.05, less
than the statisticalM error bar. The probability of ) [ 0 is
"
still over 99%. When we Ðtted with the smaller systematic
error in the opposite direction (i.e., high-redshift supernovae
discovered to be 0.03 mag fainter than measured), we Ðnd
(Ðt I) only a 0.04 shift in )flat from Ðt C.
M
CONCLUSIONS AND DISCUSSION
The conÐdence regions of Figure 7 and the residual plot
of Figure 2b lead to several striking implications. First, the
data are strongly inconsistent with the " \ 0, Ñat universe
model (indicated with a circle) that has been the theoretically favored cosmology. If the simplest inÑationary theories are correct and the universe is spatially Ñat, then the
supernova data imply that there is a signiÐcant, positive
cosmological constant. Thus the universe may be Ñat or
there may be little or no cosmological constant, but the
data are not consistent with both possibilities simultaneously. This is the most unambiguous result of the current
data set.
Second, this data set directly addresses the age of the
universe relative to the Hubble time, H~1. Figure 9 shows
that the ) -) conÐdence regions are0 almost parallel to
"
contours ofMconstant
age. For any value of the Hubble constant less than H \ 70 km s~1 Mpc~1, the implied age of
0
the universe is greater
than 13 Gyr, allowing enough time
for the oldest stars in globular clusters to evolve (Chaboyer
et al. 1998 ; Gratton et al. 1997). Integrating over ) and
M is
) , the best-Ðt value of the age in Hubble-time units
"
H t \ 0.93`0.06 or, equivalently, t \ 14.5`1.0(0.63/h)
0 0 The age~0.06
0 in a Ñat
~1.0
Gyr.
would be somewhat larger
universe :
H tflat \ 0.96`0.09 or, equivalently, tflat \ 14.9`1.4(0.63/h)
0 0
~0.07
0
~1.1
Gyr.
Third, even if the universe is not Ñat, the conÐdence
regions of Figure 7 suggest that the cosmological constant
is a signiÐcant constituent of the energy density of the universe. The best-Ðt model (the center of the shaded contours)
indicates that the energy density in the cosmological constant is D0.5 more than that in the form of mass energy
density. All of the alternative Ðts listed in Table 3 indicate a
positive cosmological constant with conÐdence levels of
order 99%, even with the systematic uncertainty included in
the Ðt or with a clumped-matter metric.
Given the potentially revolutionary nature of this third
conclusion, it is important to reexamine the evidence carefully to Ðnd possible loopholes. None of the identiÐed
582
PERLMUTTER ET AL.
Vol. 517
TABLE 4
SUMMARY OF UNCERTAINTIES AND CROSS-CHECKS
A. CALCULATED IDENTIFIED UNCERTAINTIES
Uncertainty on () flat, )flat) \ (0.28, 0.72)a
M
"
Source of Uncertainty
Statistical uncertainties (see ° 5) :
High-redshift supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Low-redshift supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Total . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Systematic uncertainties from identiÐed entities/processes :
Dust that reddens, i.e., R (z \ 0.5) \ 2R (z \ 0) (see ° 4.1.2) . . . . . . . . . . . . . .
B
B
Malmquist bias di†erence (see ° 4.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
K-correction uncertainty (see °° 2 and 3) including zero points . . . . . . . . . . .
Evolution of average SN Ia progenitor metallicity
a†ecting rest-frame B spectral features (see ° 4.4) . . . . . . . . . . . . . . . . . . . . . . . . .
Total . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.05
0.065
0.085
\0.03
\0.04
\0.025
\0.01
0.05
B. UNCERTAINTIES NOT CALCULATED
Proposed/Theoretical Sources of Systematic Uncertainties
Bounds and Tests (see text)
Evolving gray dust, i.e., R (z \ 0.5) \ 2R (z \ 0) (see °° 4.4 and 4.1.3) . . . . . .
B
B
Clumpy gray dust (see ° 5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SN Ia evolution e†ects (see ° 4.4)b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Shifting distribution of progenitor mass, metallicity, C/O ratio . . . . . . . . . . . .
Test with º3-Ðlter color measurements
Would increase SN mag residual dispersion with z
Test that spectra match on appropriate date for all z
Compare low- and high-redshift light-curve rise-times, and light-curve
timescales before and after maximum. Test width-luminosity relation
for low-redshift supernovae across wide range of environments.
Compare low- and high-redshift subsets from ellipticals/spirals,
cores/outskirts, etc.
C. CROSS CHECKS
Sensitivity to (see ° 4.5)
Width-luminosity relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
NonÈSN Ia contamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Galactic extinction model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gravitational lensing by clumped mass (see ° 4.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
\
*
)M,"flat
\0.03
\0.05
\0.04
\0.06
a For the redshift distribution of supernovae in this work, uncertainties in )flat correspond approximately to a factor of 1.3 times uncertainties in the
M," the small di†erences between the positive and negative error bars ; see
relative supernova magnitudes. For ease of comparisons, this table does not distinguish
Table 3 for these.
b The comparison of low- and high-redshift light-curve rise times discussed in ° 4.4 theoretically limits evolutionary changes in the zero-point of the
[ 0.13.
light-curve width-luminosity relation to less than D 0.1 mag, i.e., *
)M,"flat
sources of statistical and systematic uncertainty described
in the previous sections could account for the data in a
" \ 0 universe. If the universe does in fact have zero cosmological constant, then some additional physical e†ect or
““ conspiracy ÏÏ of statistical e†ects must be operativeÈand
must make the high-redshift supernovae appear almost 0.15
mag (D15% in Ñux) fainter than the low-redshift supernovae. At this stage in the study of SNe Ia, we consider this
unlikely but not impossible. For example, as mentioned
above, some carefully constructed smooth distribution of
large-grainÈsized gray dust that evolves similarly for elliptical and spiral galaxies could evade our current tests. Also,
the full data set of well-studied SNe Ia is still relatively
small, particularly at low redshifts, and we would like to see
a more extensive study of SNe Ia in many di†erent hostgalaxy environments before we consider all plausible loopholes (including those listed in Table 4B) to be closed.
Many of these residual concerns about the measurement
can be addressed with new studies of low-redshift supernovae. Larger samples of well-studied low-redshift supernovae will permit detailed analyses of statistically
signiÐcant SN Ia subsamples in di†ering host environments.
For example, the width-luminosity relation can be checked
and compared for supernovae in elliptical host galaxies, in
the cores of spiral galaxies, and in the outskirts of spiral
galaxies. This comparison can mimic the e†ects of Ðnding
high-redshift supernovae with a range of progenitor ages,
metallicities, and so on. So far, the results of such studies
with small statistics has not shown any di†erence in widthluminosity relation for this range of environments. These
empirical tests of the SNe Ia can also be complemented by
better theoretical models. As the data sets improve, we can
expect to learn more about the physics of SN Ia explosions
and their dependence on the progenitor environment,
strengthening the conÐdence in the empirical calibrations.
Finally, new, well-controlled, digital searches for SNe Ia at
low redshift will also be able to further reduce the uncertainties due to systematics such as Malmquist bias.
6.1. Comparison with Previous Results
A comparison with the Ðrst supernova measurement of
the cosmological parameters in P97 highlights an important aspect of the current measurement. As discussed in ° 3,
the P97 measurement was strongly skewed by SN 1994H,
one of the two supernovae that are clear statistical outliers
from the current 42 supernova distribution. If SN 1994H
had not been included in the P97 sample, then the cosmological measurements would have agreed within the 1 p
No. 2, 1999
) AND " FROM 42 HIGH-REDSHIFT SUPERNOVAE
error bars with the current result. (The small changes in the
K-corrections discussed in ° 2 are not a signiÐcant factor in
arriving at this agreement.) With the small P97 sample size
of seven supernovae (only Ðve of which were used in the P97
width-corrected analysis) and somewhat larger measurement uncertainties, it was not possible to distinguish SN
1994H as the statistical outlier. It is only with the much
larger current sample size that it is easy to distinguish such
outliers on a graph such as Figure 2c.
The fact that there are any outliers at all raises one cautionary Ñag for the current measurement. Although neither
of the current two outliers is a clearly aberrant SN Ia (one
has no SN Ia spectral conÐrmation, and the other has a
relatively poor constraint on host-galaxy extinction), we are
watching carefully for such aberrant events in future lowand high-redshift data sets. Ideally, the one-parameter
width-luminosity relationship for SNe Ia would completely
account for every single well-studied SN Ia event. This is
not a requirement for a robust measurement, but any exceptions that are discovered would provide an indicator of
as-yet-undetected parameters within the main SN Ia distribution.
Our Ðrst presentation of the cosmological parameter
measurement (Perlmutter et al. 1997f), based on 40 of the
current 42 high-redshift supernovae, found the same basic
results as the current analysis : A Ñat universe was shown to
require a cosmological constant, and only a small region of
lowÈmass-density parameter space with all the systematic
uncertainty included could allow for " \ 0. (Fit M of
Figure 5f still shows almost the same conÐdence region,
with the same analysis approach.) The current conÐdence
region of Figure 7 has changed very little from the corresponding conÐdence region of Perlmutter et al. (1997f), but
since most of the uncertainties in the low-redshift data set
are now included in the statistical error, the remaining systematic error is now a small part of the error budget.
The more recent analyses of 16 high-redshift supernovae
by Riess et al. (1998) also show a very similar ) -) con" uniÐdence region. The best Ðts for mass density in Ma Ñat
verse are )flat \ 0.28 ^ 0.10 or )flat \ 0.16 ^ 0.09 for the
M analyses of their Mnine independent, welltwo alternative
observed, spectroscopically conÐrmed supernovae. The best
Ðts for the age of the universe for these analyses are H t \
0 0 et
0.90`0.07 and H t \ 0.98`0.07. To Ðrst order, the Reiss
~0.05
0
0
~0.05
al. result provides an important independent cross-check
for all three conclusions discussed above, since it was based
on a separate high-redshift supernova search and analysis
chain (see Schmidt et al. 1998). One caveat, however, is that
their ) -) conÐdence region result cannot be directly
M to
" ours to check for independent consistency,
compared
because the low-redshift supernova data sets are not independent : a large fraction of these supernovae with the
highest weight in both analyses are from the Calan/Tololo
Supernova Survey (which provided many well-measured
supernovae that were far enough into the Hubble Ñow so
that their peculiar velocities added negligible redshiftuncertainty). Moreover, two of the 16 high-redshift supernovae included in the Reiss et al. conÐdence-region analyses
were from our sample of 42 Supernova Cosmology Project
supernovae ; Riess et al. included them with an alternative
analysis technique applied to a subset of our photometry
results. (In particular, their result uses the highest redshift
supernova from our 42 supernova sample, which has strong
weight in our analysis due to the excellent HST photometry.) Finally, although the analysis techniques are mostly
583
independent, the K-corrections are based on the same
approach (Nugent et al. 1998) discussed above.
6.2. Comparison with Complementary Constraints on
) and )
M
"
SigniÐcant progress is being made in the measurement of
the cosmological parameters using complementary techniques that are sensitive to di†erent linear combinations of
) and ) , and have di†erent potential systematics or
M
"
model
dependencies.
Dynamical methods, for example, are
particularly sensitive to ) , since ) a†ects dynamics only
M
"
weakly. Since there is evidence that dynamical estimates of
) depend on scale, the most appropriate measures to
M
compare with our result are those obtained on large scales.
From the abundanceÈindeed, the mere existenceÈof rich
clusters at high redshift, Bahcall & Fan (1998) Ðnd ) \
M
0.2`0.3 (95% conÐdence). The Canadian Network for
~0.1
Observational Cosmology collaboration (Carlberg et al.
1996 ; Carlberg et al. 1998) applies evolution-corrected
mass-to-light ratios determined from virial mass estimates
of X-ray clusters to the luminosity density of the universe
and Ðnds ) \ 0.17 ^ 0.07 for ) \ 0 (D90% conÐdence),
M
with small changes
in these results" for di†erent values of )
"
(cf. Carlberg 1997). Detailed studies of the peculiar velocities of galaxies (e.g., Willick et al. 1997 ; Willick & Strauss
1998 ; Riess et al. 1997b ; but see Sigad et al. 1998) are now
giving estimates of b \ )0.6/b
B 0.45 ^ 0.11 (95%
IRASof density contrast in
conÐdence),14 where b is theM ratio
galaxies compared to that in all matter. Under the simplest
assumption of no large-scale biasing for IRAS galaxies,
b \ 1, these results give ) B 0.26 ^ 0.11 (95% conÐdence),
M dynamical estimatesÈand with
in agreement with the other
our supernova results for a Ñat cosmology.
A form of the angular-size distance cosmological test has
been developed in a series of papers (see Guerra & Daly
1998 and references therein) and implemented for a sample
of 14 radio galaxies by Daly, Guerra, & Wan (1998). The
method uses the mean observed separation of the radio
lobes compared to a canonical maximum lobe sizeÈ
calculated from the inferred magnetic Ðeld strength, lobe
propagation velocity, and lobe widthÈas a calibrated standard ruler. The conÐdence region in the ) -) plane
M with
"
shown in Daly et al. (1998) is in broad agreement
the
SN Ia results we report ; they Ðnd ) \ 0.2`0.3 (68%
M
~0.2
conÐdence) for a Ñat cosmology.
Quasi-stellarÈobject gravitational lensing statistics are
dependent on both volume and relative distances and thus
are more sensitive to ) . Using gravitational lensing statistics, Kochanek (1996) "Ðnds ) \ 0.66 (at 95% conÐdence
for ) ] ) \ 1) and ) [ "0.15. Falco, Kochanek, &
M (1998)
" obtained further
M
Munoz
information on the redshift
distribution of radio sources, which allows calculation of
the absolute lensing probability for both optical and radio
lenses. Formally, their 90% conÐdence levels in the ) -)
M "
plane have no overlap with those we report here. However,
as Falco et al. (1998) discuss, these results do depend on the
choice of galaxy subtype luminosity functions in the lens
models. Chiba & Yoshii (1999) emphasized this point,
reporting an analysis with E/S0 luminosity functions that
yielded a best-Ðt mass density in a Ñat cosmology of )flat \
M
0.3`0.2, in agreement with our SN Ia results.
~0.1
14 This is an error-weighted mean of Willick et al. (1997) and Riess et al.
(1997b), with optical results converted to equivalent IRAS results using
b /b
\ 1.20 ^ 0.05 from Oliver et al. (1996).
Opt IRAS
584
PERLMUTTER ET AL.
Vol. 517
Several papers have emphasized that upcoming balloon
and satellite studies of the cosmic background radiation
(CBR) should provide a good measurement of the sum of
the energy densities, ) ] ) , and thus provide almost
M
"
orthogonal information to the supernova measurements
(White 1998 ; Tegmark et al. 1999). In particular, the position of the Ðrst acoustic peak in the CBR power spectrum is
sensitive to this combination of the cosmological parameters. The current results, while not conclusive, are already
somewhat inconsistent with overclosed () ] ) ? 1) cosM
"
mologies and ““ near-empty ÏÏ () ] ) [ 0.4) cosmologies
M
"
and may exclude the upper right and lower left regions of
Figure 7 (see, e.g., Lineweaver 1998 ; Efstathiou et al. 1998).
range 0.2 [ ) [ 0.4. However, all the cosmological
M
models shown in Figure 10 still require that the initial conditions for the new energy density be tuned with extreme
precision to reach their current-day values. Zlatev, Wang, &
Steinhardt (1999) have shown that some theories with timevarying w naturally channel the new energy density term to
““ track ÏÏ the matter term as the universe expands, leadingÈ
without coincidencesÈto values of an e†ective vacuum
energy density today that are comparable to the mass
energy density. These models require w Z [0.8 at all times
up to the present for ) º 0.2. The supernova data set
M
presented here and future complementary data sets will
allow us to explore these possibilities.
6.3. Cosmological Implications
If in fact the universe has a dominant energy contribution
from a cosmological constant, there are two coincidences
that must be addressed in future cosmological theories.
First, a cosmological constant in the range shown in Figure
7 corresponds to a very small energy density relative to the
vacuumÈenergy-density scale of particle physics energy zero
points (see Carroll, Press, & Turner 1992 for a discussion of
this point). Previously, this had been seen as an argument
for a zero cosmological constant, since presumably some
symmetry of the particle physics model is causing cancellations of this vacuum energy density. Now it would be
necessary to explain how this value comes to be so small,
yet nonzero.
Second, there is the coincidence that the cosmological
constant value is comparable to the current mass-energy
density. As the universe expands, the matter energy density
falls as the third power of the scale, while the cosmological
constant remains unchanged. One therefore would require
initial conditions in which the ratio of densities is a special,
inÐnitesimal value of order 10~100 in order for the two
densities to coincide today. [The cross-over between massdominated and "-dominated energy density occurred at
z B 0.37 for a Ñat ) B 0.28 universe, whereas the crossM
over between deceleration
and acceleration occurred when
(1 ] z)3) /2 \ ) , that is at z B 0.73. This was approxM SN 1997G
"
imately when
exploded, over 6 ] 109 yr ago.]
It has been suggested that these cosmological coincidences could be explained if the magnitude-redshift relation we Ðnd for SNe Ia is due not to a cosmological
constant, but rather to a di†erent, previously unknown
physical entity that contributes to the universeÏs total
energy density (see, e.g., Steinhardt 1996 ; Turner & White
1997 ; Caldwell, Dave, & Steinhardt 1998). Such an entity
can lead to a di†erent expansion history than the cosmological constant does, because it can have a di†erent relation (““ equation of state ÏÏ) between its density o and pressure
p than that of the cosmological constant, p /o \ [1. We
" "
can obtain constraints on this equation-of-state
ratio,
w 4 p/o, and check for consistency with alternative theories
(including the cosmological constant with w \ [1) by
Ðtting the alternative expansion histories to data ; White
(1998) has discussed such constraints from earlier supernova and CBR results. In Figure 10, we update these constraints for our current supernova data set for the simplest
case of a Ñat universe and an equation of state that does not
vary in time (see Garnavich et al. 1998b for comparison
with their high-redshift supernova data set and Aldering et
al. 1998 for time-varying equations of state Ðtted to our
data set). In this simple case, a cosmological-constant equation of state can Ðt our data if the mass density is in the
The observations described in this paper were primarily
obtained as visiting/guest astronomers at the CTIO 4 m
telescope, operated by the National Optical Astronomy
Observatory under contract to the NSF ; the Keck I and II
10 m telescopes of the California Association for Research
in Astronomy ; the WIYN telescope ; the ESO 3.6 m telescope ; the INT and WHT, operated by the Royal Greenwich Observatory at the Spanish Observatorio del Roque
de los Muchachos of the Instituto de Astrof• sica de Canarias ; the HST ; and the Nordic Optical 2.5 m telescope. We
thank the dedicated sta†s of these observatories for their
excellent assistance in pursuit of this project. In particular,
Dianne Harmer, Paul Smith, and Daryl Willmarth were
extraordinarily helpful as the WIYN queue observers. We
thank Gary Bernstein and Tony Tyson for developing and
supporting the Big Throughput Camera at the CTIO 4 m ;
this wide-Ðeld camera was important in the discovery of
many of the high-redshift supernovae. David Schlegel,
Doug Finkbeiner, and Marc Davis provided early access to
and helpful discussions concerning their models of Galactic
extinction. Megan Donahue contributed serendipitous
HST observations of SN 1996cl. We thank Daniel Holz and
Peter HoŽÑich for helpful discussions. The larger computations described in this paper were performed at the U.S.
Department of EnergyÏs National Energy Research Science
Computing Center. This work was supported in part by the
FIG. 10.ÈBest-Ðt 68%, 90%, 95%, and 99% conÐdence regions in the
) -w plane for an additional energy density component, ) , characterized
M
by an equation of state w \ p/o. (If this energy densityw component is
EinsteinÏs cosmological constant, ", then the equation of state is w \
p /o \ [1.) The Ðt is for the supernova subset of our primary analysis, Ðt
"
C," constrained
to a Ñat cosmology () ] ) \ 1). The two variables M
w
B
and a are included in the Ðt and thenMintegrated
over to obtain the twodimensional probability distribution shown.
No. 2, 1999
) AND " FROM 42 HIGH-REDSHIFT SUPERNOVAE
Physics Division, E. O. Lawrence Berkeley National Laboratory of the U.S. Department of Energy under Contract
DE-AC03-76SF000098, and by the NSFÏs Center for Particle Astrophysics, University of California, Berkeley under
grant ADT-88909616. A. V. F. acknowledges the support of
585
NSF grant AST 94-17213, and A. G. acknowledges the
support of the Swedish Natural Science Research Council.
The France-Berkeley Fund and the Stockholm-Berkeley
Fund provided additional collaboration support.
APPENDIX
EXTINCTION CORRECTION USING A BAYESIAN PRIOR
BayesÏs theorem provides a means of estimating the a posteriori probability distribution, P(A o A ), of a variable A given a
m
measurement of its value, A , along with a priori information, P(A), about what values are likely :
m
P(A o A)P(A)
m
P(A o A ) \
.
(A1)
m
/ P(A o A)P(A)dA
m
In practice P(A) often is not well known but must be estimated from sketchy, and possibly biased, data. For our purposes
here we wish to distinguish between the true probability distribution, P(A), and its estimated or assumed distribution, often
called the Bayesian prior, which we denote as P(A). RPK96 present a Bayesian method of correcting SNe Ia for host galaxy
extinction. For P(A) they assume a one-sided Gaussian function of extinction, G(A), with dispersion p \ 1 magnitude :
G
q J2/(np2G ) e~A2@2p2G for A º 0 ,
(A2)
P(A) \ G(A) 4 r
for A \ 0 ,
s0
which reÑects the fact that dust can only redden and dim the light from a supernova. The probability distribution of the
measured extinction, A , is an ordinary Gaussian with dispersion p , i.e., the measurement uncertainty. RPK96 choose the
m o A ) as their best estimate of the extinction
m for each supernova :
most probable value of P(A
m
q A p2
t m G
for A [ 0 ,
m
(A3)
AΠ\ mode [P(A o A )] \ r p2 ] p2
G
m
m
t G
for A ¹ 0 .
s0
m
Although this method provides the best estimate of the extinction correction for an individual supernova, provided
P(A) \ P(A), once measurement uncertainties are considered, its application to an ensemble of SNe Ia can result in a biased
estimate of the ensemble average extinction whether or not P(A) \ P(A). An extreme case that illustrates this point is where
the true extinction is zero for all supernovae, i.e., P(A) is a delta function at zero. In this case, a measured value of
E(B [ V ) \ 0 (too blue) results in an extinction estimate of AŒ \ 0, while a measured value with E(B [ V ) º 0 results in an
G estimates will be
extinction estimate AΠ[ 0. The ensemble mean of these extinction
G
p2
p
G
,
(A4)
SAΠT \ m
G
J2n p2G ] p2m
rather than 0 as it should be. (This result is changed only slightly if the smaller uncertainties assigned to the least extincted
SNe Ia are incorporated into a weighted average.)
The amount of this bias is dependent on the size of the extinction-measurement uncertainties, p \ R p
. For our
m
B E(B~V)
sample of high-redshift supernovae, typical values of this uncertainty are p D 0.5, whereas for the low-redshift
supernovae,
m 0, while the one-sided prior, G(A), of equation
p D 0.07. Thus, if the true extinction distribution is a delta function at A \
m
(A2) is used, the bias in SAΠT is about 0.13 mag in the sense that the high-redshift supernovae would be overcorrected for
G amount of bias depends on the details of the data set (e.g., color uncertainty and relative
extinction. Clearly, the exact
weighting), the true distribution P(A), and the choice of prior P(A). This is a worst-case estimate, since we believe that the true
extinction distribution is more likely to have some tail of events with extinction. Indeed, numerical calculations using a
one-sided Gaussian for the true distribution, P(A), show that the amount of bias decreases as the Gaussian width increases
away from a delta function, crosses zero when P(A) is still much narrower than P(A), and then increases with opposite sign.
One might use the mean of P(A o A ) instead of the mode in equation (A3), since the bias then vanishes if P(A) \ P(A) ;
however, this mean-calculated bias ismeven more sensitive to P(A) D P(A) than the mode-calculated bias.
We have only used conservative priors (which are somewhat broader than the true distribution, as discussed in ° 4.1) ;
however, it is instructive to consider the bias that results for a less conservative choice of prior. For example, an extinction
distribution with only half of the supernovae distributed in a one-sided Gaussian and half in a delta function at zero
extinction is closer to the simulations given by Hatano et al. (1998). The presence of the delta-function component in this less
conservative prior assigns zero extinction to the vast majority of supernovae and thus cannot produce a bias even with
di†erent uncertainties at low and high redshift. This will lower the overall bias, but it will also assign zero extinction to many
more supernovae than assumed in the prior, in typical cases in which the measurement uncertainty is not signiÐcantly smaller
than the true extinction distribution. A restrictive prior, i.e., one that is actually narrower than the true distribution, can even
lead to a bias in the opposite direction from a conservative prior.
A
B
586
PERLMUTTER ET AL.
It is clear from BayesÏs theorem itself that the correct procedure for determining the maximum-likelihood extinction, A3 , of
an ensemble of supernovae is to Ðrst calculate the a posteriori probability distribution for the ensemble :
P(A) ; P(A o A)
mi
,
(A5)
P(A o MA N) \
mi
/ P(A) ; P(A o A)dA
mi
and then take the most probable value of P(A o MA N) for A3 . For the above example of no reddening, this returns the correct
mi
value of A3 \ 0.
In Ðtting the cosmological parameters generally one is not quite as interested in the ensemble extinction as in the combined
impact of individual extinctions. In this case P(A o MA N) must be combined with other sources of uncertainty for each
i
supernova in a maximum-likelihood Ðt, or the use of a mBayesian prior must be abandoned. In the former case a s2 Ðt is no
longer appropriate, since the individual P(A o MA N)Ïs are strongly non-Gaussian. Use of a Gaussian uncertainty for AŒ based
mi
G
on the second moment of P(A o MA N) may introduce additional biases.
mi
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